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Graduate Studies
5-2010
Pulsed-Laser Excited Photothermal Study of
Glasses and Nanoliter Cylindrical Sample Cell
Based on Thermal Lens Spectroscopy
Prakash Raj Joshi
Utah State University
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PULSED-LASER EXCITED PHOTOTHERMAL STUDY OF GLASSES AND
NANOLITER CYLINDRICAL SAMPLE CELL BASED ON THERMAL LENS
SPECTROSCOPY
by
Prakash Raj Joshi
A dissertation submitted in partial fulfillment
of the requirements for the degree
of
DOCTOR OF PHILOSOPHY
in
Chemistry
Approved:
-------------------------Dr. Stephen E. Bialkowski
Major Professor
---------------------Dr. Robert S. Brown
Committee Member
-------------------------Dr. Alexander I. Boldyrev
Committee Member
---------------------Dr. T. C. Shen
Committee Member
-------------------------Dr. David Farrelly
Committee Member
----------------------Dr. Byron R. Burnham
Dean of Graduate Studies
UTAH STATE UNIVERSITY
Logan, Utah
2010
ii
ABSTRACT
Pulsed-laser Excited Photothermal Study of Glasses and Nanoliter Cylindrical Sample
Cell Based on Thermal Lens Spectroscopy
by
Prakash Raj Joshi, Doctor of Philosophy
Utah State University, 2010
Major Professor: Stephen E. Bialkowski
Department: Chemistry and Biochemistry
The research in this dissertation presents Pulsed-Laser Excited photothermal
studies of optical glasses and cylindrical sample cell.
First, a study of a photothermal lens experiment and the finite element analysis
modeling for commercial colored glass filters is done. The ideal situation of a semiinfinite cylinder approximate model used to describe the photothermal lens experiment
requires the boundary condition that there is no transfer of heat from the glass to
surrounding when the glass is excited with a laser. The finite element analysis modeling
for photothermal signal with coupling heat with surrounding shows the thermal heat
transfer between the glass surface and the coupling fluid. This work shows that the
problem can be resolved by using pulsed laser excitation where the signal decay is faster
than the heat diffusion to the surrounding, and finite element analysis modeling to correct
the likely deviation from semi-infinite cylinder approximate models.
iii
Second, finite element analysis modeling of a photothermal lens signal also
shows that there are slow and fast components of signals, which are detected by using a
fast response detector and is explained to be due to the axial and radial transfer of heat. A
semi-analytical theoretical description of the mode-mismatched continuous and pulsedlaser excitation thermal lens effect that accounts for heat coupling both within the sample
and out to the surrounding is presented. The results are compared with the finite element
analysis solution and found to be an excellent agreement. The analytical model is then
used to quantify the effect of the heat transfer from the sample surface to the air coupling
fluid on the thermal lens signal. The results showed that the air signal contribution to the
total photothermal lens signal is significant in many cases.
Third, surface deformation phenomena are quite common when glasses are excited
by laser. Finite element analysis modeling of a surface deformation phenomenon is done.
A thermal lens reflection experiment is carried out and results are compared with
modeling. The effect of coupling fluid on sample is taken in to account to make more
accurate measurement of thermophysical properties of solid sample.
Fourth, a novel apparatus for performing photothermal lens spectroscopy is
described which uses a low-volume cylindrical sample cell with a pulsed excitation laser.
Finite element analysis modeling is used to examine the temperature profile and the
photothermal signal. The result of finite element analysis is compared with the
experimental result. The experimental photothermal lens enhancement has been found to
be that predicted from theory within experimental error.
(176 Pages)
iv
DEDICATION
In memory of my Iza (Mom)
v
ACKNOWLEDGMENTS
First of all I would like to record my deepest gratitude to my major advisor,
Professor Stephen E. Bialkowski, for his supervision, advice, and guidance from the very
early stage of this research as well as giving me extraordinary experiences throughout the
work. Above all and the most needed, he provided me unflinching encouragement and
support in various ways. His truly scientist intuition has made him as a constant oasis of
ideas and passions in science, which exceptionally inspire and enrich my growth as a
student, a researcher, and a scientist want-to-be. I am indebted to him more than he
knows.
I gratefully acknowledge all of my advisory committee members, Dr. Robert S.
Brown, Dr. Alexander I. Boldyrev, Dr. David Farrelly, and Dr. T. C. Shen, for their time,
advice, and supervision during my education at Utah State University. I am also grateful
to my former advisory committee member, Dr. Philip Silva, for his guidance during his
stay at Utah State University.
I would especially thank Dr. Brown for his suggestions and readiness to help
during my education at USU whenever I needed it.
There are no words to express my appreciation to my wife, Paban, whose
dedication, love, and persistent confidence in me has taken the load off my shoulder. I
owe her for being unselfish, loving, and such a caring mom of our beautiful kids
Vinayak, Gargee, and Vitaraag.
vi
Many thanks go in particular to my co-graduate students in the Department of
Chemistry and Biochemistry, especially to Oluwatosin O. Dada and Mark Erupe for their
companionship and nice time we spent together.
I would like to thank the USU Department of Chemistry and Biochemistry for
giving me an opportunity to pursue my graduate study and support in the form of a
teaching assistantship and research assistantship through 2005 to 2010.
Finally, I would like to thank everybody who was important to the successful
realization of this dissertation, as well as expressing my apology that I could not mention
each one personally.
Prakash Raj Joshi
vii
CONTENTS
Page
ABSTRACT...……………………………………………………………………………..ii
ACKNOWLEDGMENTS…………………………………………………………………v
LIST OF TABLES………………………………………………………………………..xi
LIST OF FIGURES……………………………………………………………………...xii
LIST OF ABBREVIATIONS..………………………………………………………….xvi
CHAPTER
1. INTRODUCTION ………………………………………………………………...1
GLASSES …………………………………………………………………4
PHOTOTHERMAL SPCTROSCOPY. …………………………………...6
FINITE ELEMENT ANALYSIS MODELING OF
PHOTOTHERMAL PHENOMNON …………………….........................9
ORGANINIZATION OF THE REMAINING CHAPTERS …………....11
REFERENCES …………………………………………………………..13
2. PULSED LASER EXCITED PHOTOTHERMAL LENS
SPECTROMETRY OF CdSXSe1-X DOPED SILICA GLASSES ……………….15
ABSTRACT ……………………………………………………………..15
INTRODUCTION …………………………………………………….…16
THEORY ………………………………………………………………...18
EXPERIMENTAL …………………………………………………….....24
Thermal Lensing Apparatus …………………………………….….24
Samples ………………………………………………………….....25
Finite Element Analysis …………………………………………....26
RESULTS AND DISCUSSION …………………………………….…...28
CONCLUSION …………………………………………………………..38
REFERENCES ……………………………………………………..........38
viii
3. DETECTION OF FAST AND SLOW SIGNAL DECAY COMPONENT
IN GLASS BY PHOTOTHERMAL EXPERIMENT USING FAST
RESPONSE DETECTOR...............……………………………………………...42
ABSTRACT……………………………………………………………...42
INTRODUCTION ……………………………………………………....42
THEORY ……………………………………………………………...…45
MATERIALS AND METHODS………………………………………...48
Experimental apparatus ……………………………………………..48
Sample……………………………………………………………....49
Modeling ……………………………………………………………49
RESULTS AND DISCUSSION…………………………………………..51
CONCLUSION. …………………………………………………………..56
REFERENCES …………………………………………………………...58
4. MEASUREMENT OF SURFACE DEFORMATION OF
GLASS BY PULSED LASER EXCITED PHOTOTHERMAL
REFLECTION LENS EFFECT……………………………………………….....60
ABSTRACT ……………………………………………………………...60
INTRODUCTION ………………………………………………………..60
THEORY …………………………………………………………………63
Initial surface heating.. ……………………………………………...65
Time-dependent temperature and thermoelastic displacement ……..66
Heat transfer with coupling fluid ......……………………………….67
Photothermal reflection lens ………………………………………..68
Effects of coupling fluid refractive index change …………………..68
MATERIALS AND METHODS………………………………………...69
Experimental apparatus ……………………………………………..69
Sample ..………………………………………………………….…...70
Modeling ..…………………………………………………….……...71
RESULTS AND DISCUSSION ………………………………………....72
CONCLUSION …………………………………………………………..79
RRFERENCES …………………………………………………………..79
ix
5.
PHOTOTHERMAL LENS SPECTROMETRY IN
NANOLITER CYLINDRICAL SAMPLE CELLS................................………..81
ABSTRACT………………………………………………………………81
INTRODUCTION ………………………………………………………..81
THEORY ……………………………………………………………….83
Temperature change ………………………………………………...84
Photothermal lens signal ……………………………………………86
Thermal lens signal …………………………………………………87
EXPERIMENTAL ………………………………………………………89
Sample cells and Sample ………………………………………..…90
Finite Element Analysis ……………………………………….…..91
RESULTS AND DISCUSSION ………………………………….……..92
CONCLUSION …………………………………………………………..97
RRFERENCES
………………………………………………………98
6. ANALYTICAL SOLUTION FOR MODE-MISMATCHED THERMAL
LENS SPECTROSCOPY WITH SAMPLE-FLUID HEAT COUPLING ..........100
ABSTRACT……………………………………………………….........100
INTRODUCTION ………………………………………………….......100
THEORY ……….....................................................................................102
Temperature gradient ......................................................................102
Finite Element Analysis ...................................................................107
PROBE BEAM PHASE SHIFT AND THERMAL LENS
INTENSITY..............................................................................................113
CONCLUSION .........................................................................................119
REFRENCES ............................................................................................120
7. PULSED-LASER EXCITED THERMAL LENS SPECTROSCOPY WITH
SAMPLE-FLUID HEAT COUPLING ................................................................122
ABSTRACT ......................................................................................................122
INTRODUCTION.............................................................................................122
THEORY...........................................................................................................124
RESULTS AND DISCUSSION ……………………………………...............130
CONCLUSION ……………………………………………………….............137
REFERENCES..................................................................................................138
x
8. SUMMARY …..……..……........………………………………………………140
APPENDIX .....………………………………………………………………………….144
APPENDIX A: Figure Data .....……………………………………………........145
APPENDIX B: Comsol Multiphysics Files .....……………………………........146
APPENDIX C: Permission Letters .....…………………………………….........147
CURRICULUM VITAE .....…………………………………………………………….158
xi
LIST OF TABLES
Table
Page
2-1. Thermo-optical constants of optical glasses..…………………………………….. 36
6-1. Parameters used for the simulations. The thermal, optical and mechanical
properties of glasses and air.....................................................................................109
7-1. Parameters used for the simulations. The thermal, optical and mechanical
properties of glasses and air.....................................................................................129
xii
LIST OF FIGURES
Figure
Page
2-1.
A schematic drawing of a photothermal lens experimental setup. ………………24
2-2a.
Experimental thermal lens signals for 2 mm quartz cuvette with FeCp2
solution…………………………………………………………………………...28
2-2b.
Experimental thermal lens signals for 3.05 mm thick CdSxSe1-x
Doped Corning glass filter (dotted curve) and 1.45 mm thick
CdSxSe1x doped Corning glass filter (solid curve) ……………………………….29
2-3.
FEA modeling (solid line curve) and experimental signal
(Dotted curve) of the FeCp2 solution...……………………………………..…….29
2-4a. FEA modeling of photothermal lens of FeCp2 solution ………………………….30
2-4b. FEA modeling of photothermal lens of 3.05 mm thick glass filter……………….31
2-4c.
FEA modeling of photothermal lens of air around the glass filter……………….31
2-5.
Plot of semi-infinite cylinder (dashed line) and FEA modeling of
glass with (solid line) and without (dotted line) air boundary ………………….32
3-1.
A schematic drawing of a photothermal lens experimental setup ……………….49
3-2.
FEA modeling of glass surrounded by air (2D axial symmetry)………..…….…51
3-3.
FEA modeling of 3.05 mm glass surrounded by air of 40-100
micrometer (µm) pulsed-laser excitation …………………………………………52
3-4.
Variation of time constant (tc) with square of laser width (w2) ..…………..…….53
3-5.
FEA modeling of glasses of 0.50-3.05 millimeter thickness………………...…..54
3-6.
FEA modeling of photothermal signal in one millimeter glass and air……...…..54
3-7.
FEA modeling of photothermal signal in air………………………………….....55
3-8.
Experimental Photothermal signals (transmission and reflection)………....…....56
xiii
4-1.
Schematic drawing of a photothermal lens experimental setup ............................70
4-2.
FEA model of thermal expansion of 3.05mm silica glass ......................................73
4-3.
FEA model of thermal expansion of 3.05 mm silica glass surrounded by air ........73
4-4.
FEA model (plot of temperature change) of thermal expansion
of 3.05 mm silica glass............................................................................................74
4-5.
FEA model (plot of temperature change) of thermal expansion of
3.05 mm silica glass surrounded by air ...................................................................74
4-6.
FEA modeling of (surface displacement along the z direction) silica glass ..........75
4-7.
FEA modeling of (surface displacement along the z direction) silica glass
surrounded by air ...................................................................................................75
4-8.
FEA modeling of (second derivative of z displacement with respect
to r vs. axial distance) silica glass ...........................................................................76
4-9.
FEA modeling of (second derivative of z displacement with respect
to r vs axial distance) silica glass surrounded by air..............................................76
4-10.
FEA modeling of normalized relative inverse focal length versus
time of silica glass with and without air environment ...........................................77
4-11. Reflection photothermal lens signal .......................................................................77
5-1.
A schematic drawing of a photothermal lens experimental setup ..........................90
5-2.
Photothermal lens signal of standard and cylindrical sample cell
of FeCp2 in ethanol.…………………………………...................……………….93
5-3.
FEA modeling PTL signals of cylindrical sample cells of 60 to
200 µm sizes
……………………………………………………………94
5-4.
Variation of signal magnitude with sample cell radius ……………………….....94
5-5.
FEA modeling and experimental PTL signal in 120 µm
radius sample cell of FeCp2 in ethanol solution …………………………......…..95
5-6.
Variation of signal rises time with sample cell radius ……………………………95
xiv
5-7.
Variation of time constant with cell radius ……………………………....………96
5-8.
Variation of time constant with square of the cell radius ………………………..96
6-a.
Scheme of the geometric positions of the beams in a mode-mismatched
dual-beam TL experiment ....................................................................................103
6-b.
Sample geometry used for the finite element analysis modeling ..........................103
6-2.
Temperature profile in the glass sample at z  0.5 mm using the FEA
modeling, the solution considering heat flux to air, Eq. (8), and
the solution with no transfer of heat from glass to air, Eq. (10), at
different exposure times .........................................................................................110
6-3. Temperature profile along the z direction (air-glass-air) using
the FEA modeling, the solution considering air surroundings, Eqs.(8)
and (9), and the solution with no transfer of heat from glass to air,
Eq. (10), at different exposure times. r  0 was used in the simulations................110
6-4. Temperature profile along the z direction (air-glass-air) using
the FEA modeling, the solution considering air surroundings, Eqs.
(8) and (9), and the solution with no transfer of heat from glass
to air, Eq. (10), for different radial positions ..........................................................111
6-5. Radial temperature profile in the air surroundings at t  0.12 s
using the FEA modeling, the solution for the air fluid, Eq. (9), at the
sample surface in the air side, z  0 , and up to 200 m distant from there .. ...........111
6-6. Density plot using (a) the FEA and (b) the analytical approximation models.........112
6-7. Probe beam phase shift in air and in the sample calculated using
order 0 and 1 approximations as a function of time. The inset shows
the relative error between both approximations.......................................................117
6-8. Relative difference between the zero and first order approximations for
the sample phase shift at t  0.12 s as a function of the sample thickness................118
6-9. Normalized TL signal calculated using the approximations
presented and the parameters listed in Table 6-1.
The sample thickness used was l  1mm . ..................................................................118
7-1. Normalized temperature profile in the glass sample at z  0.5 mm using
the FE modeling, the solution considering heat flux to air, Eq. (6)
at different exposure times ........................................................................................130
xv
7-2. Normalized temperature profile along the z direction (air-glass)
using the FEA modeling and the solution considering air
surroundings, Eqs. (6) and (8), at different exposure
times with r = 0 used in the simulations ...…………………………………….….131
7-3. Radial temperature profile in the air surroundings at t = 1 ms
using the FEA modeling and the solution for the air fluid,
Eq. (8), at the sample surface in the air side, z  0 ……. …..………………….......132
7-4. Normalized radial temperature profile in the sample and in the air surroundings
at t  0.5ms using Eqs. (6) and (8)....…………………………….............................132
7-5. FEA and analytical solution using Eq. (10) for the normalized
inverse focal length for the 1mm thick sample. We used
 dn / dT  f  1 106 K 1 and  ds / dT s  10 106 K 1 .....................................................133
7-6. Absolute ratio between the air and sample signal, f  t f 1 f  t s 1 , as a function
of the sample thickness with  dn / dT  f  1 106 K 1 and  ds / dT s  10 106 K 1 ..…134
7-7. Absolute ratio between the air and sample signal, f  t f 1 f  t s 1 , as a
function of the  dn / dT  f  ds / dT s ratio…...............................................................136
7-8. Analytical solution using Eq. (10) for the normalized inverse
focal length as a function of the sample thickness with
 dn / dT  f  1 106 K 1 and  ds / dT s  10 106 K 1 ......................................................136
xvi
LIST OF ABBREVIATIONS
Abbreviations
Definition
AU
Spectroscopic Absorbance Unit
cw
Continuous Wave
FEA
Finite Element Analysis
FeCp2
Iron (II) Dicyclopentadine
HeNe
Helium Neon
PDE
Partial Differential Equation
TEM
Transverse Electromagnetic Mode
TL
Thermal Lens
TLS
Thermal Lens Spectrometry
CHAPTER 1
INTRODUCTION
Realizing that, it would be easy to compare the measurement results of different
laboratories if all researchers share the same standard calibration materials for
photothermal lens apparatus, our laboratory started experimenting on colored glass filters,
which could serve as such a standard instead of using liquid sample. Colored glass or
volume-absorbing neutral density filters are ubiquitous, stable over time, and easy to
handle. Thermal lens experiments in reflection mode and thermal deflection confirmed
the induced thermal expansion in the sample and the heat transfer to the coupling fluid.
After carrying out the experiments and finite element analysis (FEA) of continuous laserexcited photothermal spectrometry of CdSxSe1-x doped glasses, two problems were
realized.1
First, the colored glass and neutral density filters that we examined had anomalous
physical properties. In particular, we find that these optical glasses have a positive
temperature-dependent refractive index change coefficient. Second, heat transfer from a
glass plate excited with a laser is far from the ideal situation described by the semiinfinite cylinder approximate models that are most often used to describe the
photothermal lens experiment.
The problems have been resolved, first, by using pulsed laser excitation which
creates a temperature change on timescales short compared to heat transfer, and second
by using FEA to model heat transfer to the surroundings to take account of the likely
deviation from the semi-infinite cylinder approximate models. FEA modeling is
2
particularly useful to account for surface heat transfer that will occur in glasses and
other samples where the absorbing material is in direct contact with the coupling fluid
surroundings.
In pulsed laser excitation, the temperature change is produced almost instantly or at
least in an amount of time during which little heat has diffused to the surroundings. In
this regard, pulsed laser excitation can be advantageous for measuring absorbance.
Photothermal signals are modeled for heat transfer from glass to air after laser excitation
and the FEA model results are subsequently compared to the experimental signals. FEA
model results for colored glass filters with and without boundary conditions and the
experimental investigations of thermal lensing in colored glass filters and liquid sample
cells are presented. Thermal lens experiments and FEA modeling are presented to
investigate the temperature profile, change in refractive index of CdSxSe1-x-doped glass,
and to determine the thermo-optical coefficient, dn/dT.2
The FEA modeling of colored silica glass, of absorption coefficient 3.8 m-1with the
surrounding environment being air shows that there are slow and fast signal decay
component in glass. And time constants varies with square of the excitation beam width
according to the equation, tc = w2/4Dt, The fast and slow decay of signals are interpreted
to be due to the fast transfer of heat in axial and slow transfer of heat in radial direction of
the excitation laser beam. After realizing the fact of fast and slow components in glass,
transmission and reflection photothermal experiments were carried out using fast
response detector to detect the event. And that fast and slow decay of signal were found.
Works done in collaboration with Nelson G. C. Astrath, Luis C. Malacarne, Paulo R. B.
3
Pedreira, Marcos P. Belancon, and Mauro L. Baesso from Departamento de Física,
Universidade Estadual de Maringá, Maringá-PR, 87020-900, Brazil, have been presented.
In which a analytical theoretical description of the mode-mismatched continuous and
pulsed-laser excitation thermal lens effect, by taking into account the coupling of heat
both within the sample and out to the surroundings, is presented. The results are
compared with finite elemental analysis solution and found to be an excellent agreement.
The analytical model is then used to quantify the effect of the heat transfer from the
sample surface to the air coupling fluid on the thermal lens signal. The results showed
that the air signal contribution to the total photothermal lens signal is significant in many
cases.
When focused excitation laser beam shines on a solid material, one of the most
ordinary observed effects is the surface displacement or deformation. The absorbed
energy converts to heat, resulting in expansion and then local surface displacement of the
solid sample.
The FEA modeling shows that there is surface deformation due to pulse laser
heating of colored glass filters, and consequently the optical path length of them changed.
The modeling also shows that the extent of deformation increases with absorption
coefficient of glass. It has been found that the photothermal lens signals are affected due
to surface deformation phenomenon.
The photothermal expansion is modeled for glasses by FEA modeling with and
without heat coupling environment. Comsol Multiphysics,3.4, thermal-structural
interaction module was used to model. Time resolved reflection photothermal lens
experiment is carried out using silica glass and thermoelastic surface displacement is
4
measured. An experimentally monitored thermoelastic surface displacement result is
compared with that of modeling and are found to be very close.
A low volume cylindrical sample cell made of silica glass having an internal
diameter 240 and external diameter 1 cm, fitted in 1 cm3 steel block, for performing
photothermal lens spectroscopy with pulsed-laser excitation is described.
Experiments to verify the operation of the apparatus are performed with
dicyclopentadienyl iron (FeCp2) in ethanol. The results were compared with that of
standard conventional cell of 2 mm size. The whole sample cell volume is irradiated with
pulsed excitation laser. The photothermal lens element is formed by thermal diffusion
from the irradiated sample volume through the sample cell walls. The apparatus has been
found to work with cells designed to contain sample volumes to 361 nL. Larger and
smaller volume cells are practical. FEA modeling has been also used for the better
understanding of temperature profile and for the comparison of the experimental results.
Photothermal lens signals processed in the usual fashion are found to be relatively
linear, reproducible, and consistent with the model based on heat conduction through the
sample cell walls. The experimental photothermal lens enhancement is found to be that
predicted from theory within experimental error.
GLASSES
A glass, whether in bulk, fiber, or film form, is a non-crystalline (or amorphous)
solid. Glasses are typically brittle and often optically transparent commonly used for
windows, bottles, or eyewear. In principle, any substance can be vitrified by quenching it
5
from the liquid state, while preventing crystallization, into a solid glass. Cooling a
viscous liquid fast enough to avoid crystallization most commonly forms a glass.3-4
Most commercially available glasses prepared by melting and quenching, available
in large bulk shapes, are silicates of one type or another. Besides common silica-based
glasses, many other inorganic and organic materials may also form glasses, including
plastics (e.g., acrylic glass), carbon, metals, carbon dioxide, phosphates, borates,
chalcogenides, fluorides, germanates (glasses based on GeO2), tellurites (glasses based on
TeO2), antimonates (glasses based on Sb2O3), arsenates (glasses based on As2O3),
titanates (glasses based on TiO2), tantalates (glasses based on Ta2O5), nitrates,
carbonates, and many other substances.5-6
Some glasses that do not include silica as a major constituent may have physicochemical properties useful for their application in fibre optics and other specialized
technical applications. These include fluoride glasses (fluorozirconates,
fluoroaluminates), aluminosilicates, phosphate glasses, and chalcogenide glasses.
Glass plays an essential role in science and industry. The optical and physical
properties of glass make it suitable for applications such as flat glass, container glass,
optics and optoelectronics material, laboratory equipment, thermal insulator,
reinforcement fiber, and art. It is important to realize that some materials, which have a
very strong glassy appearance under the naked eye, or even under a laser beam, may
actually contain a fine dispersion of very minute crystals with dimensions well below 100
nm, i.e., nanocrystals.
Although most industrial glasses are based on the SiO2, many other compounds are
normally added to modify it. Cadmium sulfoselenide is the pigment currently used for
6
making red glasses. The color is due to colloidal particles of the semiconductor
cadmium sulfoselenide; photons with energies greater than the band gap of the
semiconductor undergo absorption and consequently the optical absorption spectra have a
sharp transmission cut-off. The color can be controlled by the ratio of cadmium sulfide
and cadmium selenide in the pigment, and can vary from yellow to dark red.
Semiconductor-doped glass consists of microcrystallites of II-VI semiconductor
randomly scattered in a glass host. The semiconductor crystallite can be a binary or
ternary compound giving a widely tunable band-gap. This property has lead to the
application of the glass as short wavelength cut-off filters. There has been a great deal of
interest in the glass for use as a nonlinear optical material. In order to enhance nonlinear
optical effects by increasing interaction lengths, the glass has been fabricated in both
waveguide and optical fiber form.
Glasses doped with semiconductor nanocrystals are candidates for resonant nonlinear optical materials. They show rapid response times of a few tens of ps. One such
glass has been available for many years as a sharp cut color filter; it contains a very fine
(diameter ~ 10 nm) CdSxSe1-x microcrystalline phase dispersed in the glass matrix.
PHOTOTHERMAL SPCTROSCOPY
Photothermal spectroscopy is a group of high sensitivity methods used to measure
optical absorption and thermal characteristics of a sample. The basis of photothermal
spectroscopy is a photo-induced change in the thermal state of the sample. Light energy
absorbed and not lost by subsequent emission results in sample heating. This heating
results in a temperature change as well as changes in thermodynamic parameters of the
7
sample which are related to temperature. Measurements of the temperature, pressure, or
density changes that occur due to optical absorption are ultimately the basis for
photothermal spectroscopic methods.7
Photothermal spectroscopy is more sensitive than optical absorption measured by
transmission methods. The high sensitivity of the photothermal spectroscopy methods has
led to applications for analysis of low absorbance samples.
The basic processes responsible for photothermal spectroscopy signal generation is
optical radiation, usually from a laser is used to excite a sample. Absorption of radiation
from the excitation source followed by non-radiative excited state relaxation results in
sample temperature, and density changes. The density change is primarily responsible for
the refractive index change, which can be probed by a variety of methods.7
In a generic apparatus used for photothermal spectroscopy the excitation light heats
the sample. The probe light monitors changes in the refractive index of the sample
resulting from heating. The spatial and propagation characteristics of the probe light will
be altered by the refractive index. The spatial filter selects those components of the
altered probe light that change with the samples' refractive index. The optical detector
monitors changes in the probe light power past the spatial filter. In some apparatuses, a
spatial filter and a single channel detector are combined using an image detector. Signals
generated by the photodetector are processed to enhance the signal to noise ratio.
There are several methods and techniques used in photothermal spectroscopy. Each
of these has a name indicating the specific physical effect measured. Photothermal
deflection spectroscopy, photothermal diffraction spectroscopy and photothermal lensing
spectroscopy are most commonly used techniques.7
8
The first photothermal spectroscopic method to be applied for sensitive chemical
analysis was photothermal lens spectroscopy. The photothermal lens results from optical
absorption and heating of the sample in regions localized to the extent of the excitation
source. The lens is created through the temperature dependence of the sample refractive
index. The lens usually has a negative focal length since most materials expand upon
heating and the refractive index is proportional to the density. This negative lens causes
beam divergence and the signal is detected as a time dependent decrease in power at the
center of the beam. Measurements of the change in divergence of a laser beam after
formation of the thermal lens allows determination of the absorbances of 10-7 – 10-6,
which correspond to analyte concentrations of 10-11 – 10-10 mole/L. Thus, thermal lens
spectrometry is 100 — 1000 times more sensitive than conventional spectrophotometry.
There are two main types of optical apparatus used in photothermal lens
spectroscopy. The simplest is the single-laser apparatus. The second type of apparatus
commonly used is the two-laser design. In this design one beam is used to excite the
sample, and the second beam is used to probe the resulting photothermal lens.
Thermal lens spectrometry was first reported by Gordon et al.8 It can measure very
low optical absorption coefficients of transparent samples,9 and has been used in a wide
range of applications such as trace analyses, flowing streams, flow injection, quantum
yields and chemical reaction kinetics.10 There are very good review papers on the subject.
10-12
This is a topic, which has been receiving a great deal of attention especially in the
case of some silicate, 13, 14 calcium aluminate, 15, 16 fluoride, 17 chalcogenide, and
chalcohalides glasses. Time-resolved TL technique has been shown to be a useful method
to measure the thermal diffusivity of transparent samples as well as the temperature
13, 14, 17, 18
coefficient of optical path length change.
9
Needless to say; these studies are of
great interest for glass laser designers and researchers.
The knowledge of the thermo-optical properties is essential for evaluating the
figures of merit of optical glasses. Photothermal lens spectroscopy is one of the best
methods to measure these properties.
FINITE ELEMENT ANALYSIS MODELING OF PHOTOTHERMAL PHENOMNON
The finite element analysis (FEA) is a numerical technique for finding approximate
solutions of partial differential equations (PDE) as well as of integral equations. The
solution approach is based either on eliminating the differential equation completely
(steady state problems), or rendering the PDE into an approximating system of ordinary
differential equations, which are then numerically integrated using standard techniques
such as Euler's method, Runge-Kutta, etc.
FEA is a fairly recent discipline that allows the numerical solution of governing
physical equations over complicated geometric domains. The method is regularly applied
to the structural analysis of designs with complex geometries. The part being analyzed is
divided into many small regions called "finite elements." The physical behavior within
each element is understood in concise mathematical terms. Assemblage of all elements'
behavior produces a large matrix equation, which is solved for the quantity of interest,
such as the deformation due to a maximum loading condition. Additional quantities, such
as stresses, are then computed.19
FEA is a fairly recent discipline crossing the boundaries of mathematics, physics,
and engineering and computer science. The method has wide application and enjoys
10
extensive utilization in the structural, thermal and fluid analysis areas. The finite
element method is comprised of three major phases: (1) pre-processing, in which the
analyst develops a finite element mesh to divide the subject geometry into subdomains
for mathematical analysis, and applies material properties and boundary conditions, (2)
solution, during which the program derives the governing matrix equations from the
model and solves for the primary quantities, and (3) post-processing, in which the analyst
checks the validity of the solution, examines the values of primary quantities (such as
displacements and stresses), and derives and examines additional quantities (such as
specialized stresses and error indicators).
COMSOL Multiphysics is a finite element analysis, solver and simulation
software package for various physics and engineering applications, especially coupled
phenomena. It is excellent, state-of-the-art software for the solution of many types of
partial differential equations (PDEs). Both stationary and time-dependent analysis can be
performed by numerical techniques based on the finite element method for the spatial
discretization.20
COMSOL Multiphysics also offers an extensive interface to MATLAB and its
toolboxes for a large variety of programming, preprocessing and postprocessing
possibilities. The packages are cross-platform (Windows, Mac, Linux, and UNIX.) In
addition to conventional physics-based user-interfaces, COMSOL Multiphysics also
allows for entering coupled systems of partial differential equations (PDEs). The PDEs
can be entered directly or using the so-called weak form.
The software runs FEA together with adaptive meshing and error control using a
variety of numerical solver to solve the PDEs. Models of different possible type can be
11
built in the COMSOL Multiphysics user interface. Models of heat transfer module and
the thermal structural interaction module of Comsol Multiphysics v 3.3 and 3.5 (Comsol
Inc.) in various samples are performed in this dissertation.
The analysis steps in finite element modeling are: 21
1. Defining multiphysics: Definition of the desired physics mode (heat transfer by
conduction, convection, and/or radiation) and types of solution (steady state or
transient).
2. Geometry modeling: Definition of sample geometry and materials, boundary
conditions, and heat sources and sinks.
3. Meshing: Breaking the geometry into sub-domains or element meshing.
4. Solve the model: Solving the model by choosing appropriate solver parameters.
5. Post-processing: The solution is plotted using a variety of visualization techniques.
The data from these plots can also be exported for further analysis in spreadsheet
software such as Excel.
ORGANIZATION OF THE REMAINING CHAPTERS
The rest of the chapters in this dissertation present the detailed research documents
of pulsed –laser excited photothermal lens spectroscopic method of study of glasses and
nanolitre cylindrical sample cell. FEA modeling, simulating the experimental samples,
has been done to compare and correct the experimental measurements with the help of
Comsol Multiphysics software.
Chapter 2 explains how the pulsed-laser excited photothermal lens spectrometry of
low absorbing colored glass filters overcome the problem encountered of coupling of heat
12
with the surrounding fluid while doing continues laser excited photothermal
spectroscopy. Details of thermal lens experiments and FEA modeling are presented to
investigate change in refractive index of cadmium sulfoselenium doped glass, and to
determine the thermo-optical coefficient. It turns out that the glass produce positive
refractive index change contrary to the expected negative index of refraction change.
Chapter 3 contains photothermal experiments (transmission and reflection) and the
FEA modeling details of the fast and slow signal decay component in glass. The fast and
slow decay of signals are interpreted to be due to the fast transfer of heat in axial and
slow transfer of heat in radial direction of the excitation laser beam.
In Chapter 4 details of FEA modeling of surface displacement or deformation
phenomenon observed due to photothermal heating of glass are presented. And the
effects of coupling fluids on thermal expansion and the photothermal signal are
discussed.
Experiments and the FEA modeling, to verify the operation of a nanoliter
cylindrical sample cell apparatus for performing photothermal lens spectroscopy with
pulsed-laser excitation, are discussed described in chapter 5.The apparatus has been
found to work with cells designed to contain sample volumes to 361 nL
A theoretical description of the mode-mismatched thermal lens effect by taking into
account the coupling of heat both within the sample and out to the surroundings medium
is presented in chapter 6.And the chapter 7 discuses the analytical and finite element
analysis modeling methods of the pulsed laser excited photothermal lens signal of solids
samples surrounded by air. Chapter 8 is a brief summary of the results in Chapters 2-7.
All figures data are available in Appendix A.
13
REFERENCES
1. Oluwatosin Dada, Matthew R. Jorgensen, and Stephen E. Bialkowski Applied
Spectroscopy 61, 1373-1378 (2007).
2. Prakash R. Joshi, Oluwatosin O. Dada, and Stephen E. Bialkowski Applied
Spectroscopy 63, 815 (2009).
3. Douglas, R. W. A History of Glassmaking (G T Foulis & Co Ltd., Henley-onThames, 1972).
4. Zallen, R. The Physics of Amorphous Solids (Wiley, New York, 1983).
5. Elliot, S. R., Physics of Amorphous Materials (Longman Group Ltd., London,
1984).
6. Horst Scholze, Glass – Nature, Structure, and Properties (Springer-Verlag, New
York, 1991).
7. S. E. Bialkowski, Photothermal Spectroscopy Methods for Chemical Analysis
(Wiley, New York, 1996).
8. J.P. Gordon, R.C.C. Leite, R.S. More, S.P.S. Porto, and J.R. Whinnery, J. Appl.
Phys. 36, 3 (1965).
9. J. Shen, and R.D. Snook, Anal. Proc. 26, 403 (1989).
10. R.D. Snook, and R.D. Lowe, Analyst 120, 2051 (1995).
11. K.L. Jansen, and J.M. Harris, Anal. Chem. 57, 1698 (1985).
12. H.L. Fang, and R.L. Swofford, in: D.S. Kliger (Ed.), Ultrasensitive Laser
Spectroscopy (Academic Press, New York, 1983).
13. M.L. Baesso, J. Shen, and R.D. Snook, J. Appl. Phys. 75, 3733 (1994).
14
14. M.L. Baesso, J. Shen, and R.D. Snook, Chem. Phys. Lett. 197, 255 (1992).
15. M.L. Baesso, A.C. Bento, A.A. Andrade, T. Catunda, J.A. Sampaio, and S.
Gama, J. Non-Cryst. Solids 219, 165 (1997).
16. M.L. Baesso, A.C. Bento, A.A. Andrade, T. Catunda, E. Pecoraro, L.A.O. Nunes,
J.A. Sampaio, and S. Gama, Phys. Rev. B. 57,10545 (1998).
17. S.M. Lima, T. Catunda, R. Lebullenger, A.C. Hernandes, M.L. Baesso, A.C.
Bento, and L.C.M. Miranda, Phys. Rev. B 60, 15173 (1999).
18. M.L. Baesso, A.C. Bento, A.R. Duarte, A.M. Neto, L.C. Miranda, J.A. Sampaio,
T. Catunda, S. Gama, and F.C.G. Gandra, J. Appl. Phys. 85, 8112 (1999).
19. G. Strang, and G. Fix, An Analysis of The Finite Element Method (Prentice Hall,
Englewood Cliffs, N.J., 1973).
20. Comsol Inc. Comsol multiphysics 3.3 user’s Guide.
21. Oluwatosin O Dada and Stephen E Bialkowski , Applied Spectroscopy 62, 1336
(2008).
15
CHAPTER 2
PULSED LASER EXCITED PHOTOTHERMAL LENS SPECTROMETRY OF
CdSXSe1-X DOPED SILICA GLASSESa
ABSTRACT
Experimental results for photothermal lens measurements are compared to finite
elemental analysis models for commercial colored glass filters. Finite elemental analysis
software is used to model the photothermal effect by simulating the coupling of heat both
within the sample and out to the surroundings. Modeling shows that heat transfer between
the glass surface and the air coupling fluid has a significant effect on the predicted time
dependent photothermal lens signals. For comparison with experimental signals, a simple
equation based on the finite element analysis result is proposed for accounting for the
variance of experimental data where this type of heat coupling situation occurs. The
colored glass filters are found to have positive thermo-optical coefficients. The net
positive dn/dT of CdSxSe1-x doped glass filters is considered to be the consequence of
counteracting factors: optical nonlinearity, stress-induced birefringence, and the structural
network of glass. Finite element analysis modeling results are also used to correlate
experimental measurements of different sample geometries. In particular, the glass
samples are compared to ethanol solutions of iron (II) dicylopentadiene in a sample
cuvette even though heat transfer is different for these two samples.
a
COAUTHORED BY PRAKASH RAJ JOSHI, OLUWATOSIN O. DADA, AND STEPHEN E. BIALKOWSKI.
REPRODUCED WITH PERMISSION FROM THE APPLIED SPECTROSCOPY, VOL. 63 (7), P 815 (2009)
COPYRIGHT 2009 SOCIETY FOR APPLIED SPECTROSCOPY (SEE APPENDIX C)
16
INTRODUCTION
Thermal lens spectrometry (TLS) can measure very low optical absorption
coefficients of transparent samples.1 Time resolved TLS has been shown to be a useful
method for measuring the thermal diffusivity of transparent samples as well as the
temperature coefficient of the optical path length, s, change, ds/dT.2–5 The knowledge of
the thermo-optical properties is essential for evaluating the figures of merit of optical
glasses. They can readily be used to determine the working glass conditions6,7 such as
thermal shock, thermal stress resistance,8 and thermal lens effect.6,7,9–17 Therefore,
considering the importance of determining the thermo-optical properties of laser glass
materials, a simple and accurate method to determine these properties quantitatively is of
utmost importance.
Glasses doped with semiconductor microcrystals are candidates for resonant
nonlinear optical materials. The colored glass cut-off filters that have been commercially
available for many years contain semiconductor microcrystals on the order of 10 nm and
by today’s standards these microcrystals would be called nanoparticles.18-21The
semiconductor nanoparticles are responsible for the optical absorption in these colored
glass filters. The chemical composition, size, and method of synthesis of the
nanoparticles affect the optical properties of the glass filters, most of which are high-pass
wavelength cut-off filters used in spectroscopy and other optical technologies.22
The changes in absorption of the semiconductor crystallites relative to the bulk
semiconductor lead to refractive index changes through the Kramers–Kronig
transformation. The value of χ(3) will be proportional to the reciprocal of the confinement
17
volume and will normally increase with decreasing size. Therefore, larger
nonlinearities are expected for glasses containing smaller particles and larger volume
fractions of semiconductor particles.
Pereira et al. reported the complexity of the solution to theoretical treatment of
the thermally induced bistability in semiconductor-doped glass due to the thermal lens
effect.23The photothermal lens results from optical absorption and heating of the sample
in regions localized to the extent of the excitation source. The lens is created because of
the temperature dependence of the sample refractive index. The lens usually has a
negative focal length since most materials expand upon heating and, as a first
approximation; the refractive index is simply proportional to the density. This negative
focal length lens causes beam divergence and the signal is detected as a time-dependent
decrease in power at the center of the beam.24 However, we find that the optical glasses
have a positive temperature-dependent refractive index change coefficient.
Models are used to calculate temperature changes resulting from the absorption
of excitation laser power. The combination of experiment and theory, then, allows one to
determine the temperature-dependent refractive index change, dn/dT, for the transparent
sample. Theoretical models of thermal lens effects in fluids are well established; a
number of approximations are commonly used to obtain tractable analytical results.25, 26
The ideal situation of a semi-infinite cylinder approximate model used to describe the
photothermal lens experiment assumes the boundary condition wherein no heat is
transferred from the sample to the surroundings along the dimension of the excitation
laser beam propagation.
18
In most practical situations heat is transferred from the sample to the
surroundings along the axial dimension. This is especially true when the heated sample is
in contact with air or another coupling fluid, as in the case of the glass filters used herein.
Therefore, some correction in the results from the experiment is required in order to
obtain accurate predictions of parameters derived from the photothermal effect.
The aim of this paper is to use the thermal lens method to obtain accurate
measurements of the thermo-optical properties of glasses, first by using pulsed laser
excitation, which creates a temperature change on timescales short compared to heat
transfer, and second by using finite element analysis (FEA) to model heat transfer to the
surroundings. In pulsed laser excitation, the temperature change is produced almost
instantly or at least in an amount of time during which little heat has diffused to the
surroundings. In this regard, pulsed laser excitation can be advantageous for measuring
absorbance.
Photothermal signals are modeled for heat transfer from glass to air after laser
excitation and the FEA model results are subsequently compared to the experimental
signals. FEA model results for colored glass filters with and without boundary conditions
and the experimental investigations of thermal lensing in colored glass filters and liquid
sample cells are presented.
THEORY
The thermal lens effect is caused by the deposition of heat via non-radiative decay
processes after the laser beam with Gaussian profile has been absorbed by the sample.
19
The photothermal lens signal is dependent on the spatially dependent refractive index
change produced due to the temperature change in the absorbing sample. The time- and
space-dependent temperature change is described by the differential equation for thermal
diffusion. In radial symmetry, using appropriate lasers for sample excitation, the equation
describing heat diffusion is24
q (r , z , t )

T (r , z , t )  DT  2T (r , z, t )  H
t
C p
(1)
In this equation, T(r ,z ,t) (K) is the space and time-dependent temperature
change, t (s) is time, r (m) and z (m) are radial and linear cylindrical coordinates, DT (m2
s-1) is thermal diffusion constant,  (kg m-3) and Cp (J kg-1) are density and specific heat
capacity, respectively, and qH (W m-3) is energy density.
For pulsed laser excitation on a time scale much shorter than that for thermal
diffusion, optical absorption within the sample results in sample heating that mimics the
excitation laser beam radial profile. The energy absorbed by the sample for a collimated,
short-pulsed Gaussian profile laser propagating on the z-axis is24
q H ( r , z , t )  e z
2QYH  ( 2 r 2 / w 2 )
e
 (t )
w 2
(2)
The z-axis origin is taken to be at the entrance interface of the sample. In this
equation, α (m-1) is sample exponential absorption coefficient. The first part of the energy
20
source equation is z-axis energy deposition due to sample absorption of the exponential
attenuated laser light as it traverses the sample. The second part of Eq. 2 accounts for the
radial distribution of energy produced by the Gaussian laser profile. Q (J) is laser energy,
the YH, is a heat yield parameter accounting for energy loss dues to luminescence during
sample excitation and w (m) is excitation laser beam electric field width. It is assumed
that the laser beam waste does not change over the length of the sample. The last term in
this equation is the time dependence, in this case a delta function, δ(t) which is zero
unless t = 0.
Solutions to the thermal diffusion equation are well known for the ‘infinite
cylinder’ approximation. The latter assumes that there is negligible attenuation along the
z-axis that the laser beam radius does not change, and that heat diffuses only in the radial
dimension. This approximation can be valid for thin samples that have thermally
insulated windows or are in a vacuum. For excitation pulse duration much shorter than
the thermal relaxation time, and assuming rapid excited state relaxation, Eq. 3 gives the
analytical solution to the temperature change for t ≥ 0.

2 r 2



 w2 1 2t / t  
2QYH
1
c 

e
T (r, t ) 
C p w2 (1  2t / t c )
The temperature evolution depends on the characteristic thermal diffusion time constant,
tc (s), defined by
(3)
21
tc 
w2
4 DT
(4)
DT is the thermal diffusion coefficient given by DT = k/ρCp, and k (W m-1K-1) is the
thermal conductivity of the sample.
The strength of the photothermal lens element is found from the second radial
derivative, evaluated on-axis. Integration over path length results in the inverse lens
strength27
1  dn  d 2
    2 T (r, z, t ) ds
f (t )  dT  path
dr
r 0
The theoretical treatment of thermal lens effect used the “parabolic
approximation” where the temporal evolution of the temperature profile T(r ,z, t) and,
consequently, the refractive index profile n(r, z, t) are approximated by a parabola. Thus
the second derivative with respect to radius is inversely proportional to the focal length of
the parabolic lens. dn/dT (K-1) is the temperature-dependent refractive index change, also
known as the thermal-optical coefficient. In liquids, dn/dT is usually negative, so the
photothermal lens is equivalent to a negative lens.
The change of divergence of a laser beam passing through the sample, due to the
thermal lens effect, can be measured simply by placing a pinhole over the optical detector
in the far field in the center of the diverged beam. The relative change at the detector
signal is proportional to inverse of focal length. The photothermal lens signal is related to
(5)
22
the inverse focal length. For pulsed laser excitation, the experimental photothermal
signal, S(t), is defined by24
S (t ) 
 (  )   (t )
 (t )
(6)
In Eq. 6, Φ (W) is the irradiance of the probe laser passing through a pinhole aperture
placed far away from the sample and the probe laser is focused a distance z’ (m) in front
of the sample. The signal is proportional to the inverse focal length of the lens produced
in the sample due to laser heating; S(t)≈-2z’/f(t). This is an approximation and better
signal predictions may be made with diffraction theory. Marcano et al. described the
mode-mismatched thermal lens experiment in the pulse regime.24, 28
Eqs. 3 and 5 yield a time-dependent inverse focal length under the infinite cylinder
approximation of
1
1
 dn  8lYH Q

 4
f (t )  dT  w C p (1  2t / tc ) 2
l (m) is pathlength through the sample. The magnitude of the inverse focal length directly
after excitation and assuming instantaneous temperature and refractive index change is
thus
(7)
23
1
 dn  8lYH Q


f (0)  dT  w4 C p
(8)
The dn/dT term, which can be positive or negative since it is affected by counteracting
factors. Thermal expansion decreases density and thus the refractive index due to the
greater inter-molecular spacing. Secondly, an increase in the electronic polarizability
causes the refractive index to gradually increase. In glasses, this effect is associated with
the shift for longer wavelength of the UV absorption edge. Prod’homme showed that29
2
2

dn (n0 1)(n0  2) 1 d

3

2
dT
6n0
 dT

Here  = 1/V (dV/dT)P (K-1) is the volume expansion coefficient and  is the optical
frequency polarizability coefficient. In liquids, dn/dT is usually negative as the thermal
expansion is the dominant term in Eq. (9). For glasses, dn/dT can be either positive or
negative, depending on the glass structure.30, 31 A highly expansive and loosely bound
networks as opposed to strongly bound networks presents negative and positive dn/dT,
respectively. With few exceptions, in particular, near-phase-change temperatures, most
materials expand with increased temperature. However, there are mechanisms for
increasing refractive index with temperature. Solid phase changes, expansion induced
stress, semiconductor conduction band population changes, and production of new states
may result in a positive temperature-dependent refractive index change.
(9)
24
EXPERIMENTAL
Thermal Lensing Apparatus. The diagram in Figure 2-1 illustrates the apparatus
setup for the thermal lens experiment. A Lambda Physik XeCl excimer laser pumped dye
laser operating at 490 nm was used as the excitation source. This laser delivers maximum
pulse energies of 0.44 mJ over the 20 ns pulse duration. The laser was operated with a
repetition frequency of 3.75Hz. A 632.6 nm HeNe laser (Uniphase, Model 1107P) is used
to probe the resulting photothermal lens. Collinear excitation-probe geometry was
utilized in this thermal lens set up. The distance between the sample and the photodiode
detector is optimized to satisfy the far-field paraxial approximation. Two lenses (10 cm
and 25 cm focal length) are used to focus the excitation beam in the sample after the
HeNe beam focus. The excitation source power is measured with a laser power meter.
Figure 2-1. A schematic drawing of a photothermal lens experimental setup.
25
The photothermal lens caused focusing and defocusing of the probe laser. This
is measured as a change in the power at the center of the beam. The center HeNe beam
power is measured using a pinhole and a United Detector Technology (UDT) Model PIN10DP photovoltaic photodiode detector. A 632.8 nm laser line bandpass filter is used to
prevent the transmitted dye laser beam and ambient light from being detected by the
photodiode detector. A small fraction of the probe beam is split off prior to the pinhole
aperture and a second bandpass filter/photodiode is used to monitor the probe laser power
past the sample. The photodiodes used transimpedance amplifiers. Changes in the
detected probe laser power is compensated for by using an operational amplifier divider,
the circuit of which divides the thermal lens signal by the signal proportional to the HeNe
laser power. This probe laser power compensated thermal lens signal is amplified and
electronically filtered with a Tektronix model AM-502 differential amplifier. The analog
signal is subsequently digitized with a 16-bit analog-to-digital converter board and
processed by multichannel analysis software. The latter averages several signal transients.
Multichannel averaging was performed to improve the raw photothermal lens signal
estimation precision. The photothermal lens signal was calculated from this raw data
using a simple spreadsheet program.
Samples. Standard Corning colored glass optical filters doped with cadmium
sulfoselenide microcrystals of about 10 nm are investigated as an absorbing solid
material. 5 cm x 5 cm Corning 3389 (CS3-73, 1.45, and 3.05 mm thickness) optical filters
were used for glass samples. Solutions of iron (II) dicylopentadiene (FeCp2) in ethanol
were used as the liquid standards. Linear dilution is used to obtain lower absorbance from
stock solutions of high enough absorbance to measure using a spectrophotometer in a
26
conventional 1 cm path length liquid cuvette. The sample is positioned at the focus of
the excitation beam for maximum temperature gradient, and the solid versus liquid
experiment is carried out separately at room temperature under the same conditions.
Sample absorbance was recorded with a Cary 3E UV-Visible spectrophotometer.
Finite Element Analysis. Finite Element Analysis (FEA) software provides
numerical solutions to the heat transfer equations with the realistic boundary conditions
imposed by the experimental geometry. To better understand the transient temperature
profile in the samples, FEA is used to model temperature changes. Results from the FEA
calculations are then compared to conventional analytical solutions to gage the error. The
experimental setup and the apparatus constraints are guided by the error analysis. Comsol
Multiphysics 3.4 analysis is carried out on a Compaq Presario SR1330X, AMD Athlon
XP 3200 processor using MS Windows XP.
The Comsol Multiphysics software in conduction and convection mode solves the
heat diffusion equation given as
C p

T (r , z, t )  k 2T (r , z, t )  q H (r , z, t )  C p u  T (r , z, t )
t
here u (m s-1) is the flow velocity. All other symbols are the same as those introduced
above. Note that Eq. 1 and 10 differ only by the second term on the right side of the
equation. This term can account for convection or mass flow heat transfer. Convection is
not important in glass but could be important in gases and liquids.
(10)
27
Finite element analysis modeling consists of drawing the sample geometry and
specifying material boundary conditions, heat sources, and sinks. The problems are then
solved with rough finite element definition and further refinement of elements and
domain are made. Finally, dT can be obtained either at a single time, over a time series,
or at steady state. The relative photothermal lens signal strength is found from Eq. 5 with
dn/dT set to unity. The path integral of the second radial derivative was found by using
the Comsol integration-coupling variable to integrate the second derivative function of
the temperature change.
The model solid absorbing media were small square plates, 5 cm x 5 cm x 0.305
cm, or 5 cm x 5 cm x 0.145 cm, and had the same dimensions of the glass samples used
in the experiments. Optical excitation was along the small dimension. For convenience,
glass heating was modeled with 2D axial symmetry with the origin at the center of the zaxis runs along the path length of the excitation and probe beams. We have tested this
and found that the results are equivalent to those obtained in the three dimensional
Cartesian model. However, the cylindrical model is smaller and quicker to solve. The
temperature profile of the glass was obtained by having the FEA software solve the heat
equation with and without the glass surfaces set at 0 K (assuming no convectional heat
transfer at the surfaces) and heat input along the z-axis defined by Eq. 4. In this fashion,
the temperature solution represents dT. On the other hand, room temperature is used for
temperature-dependent thermodynamic parameters of well-known library materials (air).
Our model did not consider convection because the sample is solid. The values of α, Q,
and w used were 3.8 m-1, 0.44 mJ, and 170 μm, respectively.
28
RESULTS AND DISCUSSION
Figure 2-2 shows the experimental thermal lens signals for commercial colored
glasses and a FeCp2 solution using a 490 nm pulsed excitation laser. Figure 2-2b shows
that the thin glass produces a smaller magnitude signal. The magnitude of the signal for
glass is clearly less than that for the FeCp2 solution. The conventional thermal lens signal
is expected to arise from negative focal length because most materials have a negative
dn/dT. The thermal lens signal (unitless) for a FeCp2 solution with a measured
absorbance of 0.0212 AU for 1 cm path length (Figure 2-2a) indicates a decrease in
refractive index of the sample. Figure 2-3 shows a FEA modeling and experimental
signal of the FeCp2 solution in ethanol.
a
0.200
0.180
0.160
Relative signal
0.140
0.120
0.100
0.080
0.060
0.040
0.020
0.000
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Time (s)
Figure 2-2a. Experimental thermal lens signals for 2 mm quartz cuvette with FeCp2
solution.
29
b
0.005
Relative signal
0.000
-0.005
-0.010
-0.015
-0.020
-0.025
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Time (s)
Figure 2-2b. Experimental thermal lens signals for 3.05 mm thick CdSxSe1-x doped
Corning glass filter (dotted curve) and 1.45 mm thick CdSxSe1-x doped
Corning glass filter (solid curve).
0.30
Relative signal
0.25
0.20
0.15
0.10
0.05
0.00
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Time (s)
Figure 2-3. FEA modeling (solid line curve) and experimental signal (dotted curve) of
the FeCp2 solution.
30
The thermal lens signals of Corning commercial glass filters of 3.05 mm and
1.45 mm with absorbencies 0.0117 and 0.0025 AU (Figure 2-2b) are inverted relative to
that of the FeCp2 solution, indicating a positive dn/dT. The time constant for the glass
signal is shorter than that of the FeCp2 solution due to a large DT and perhaps thermal
coupling with the surrounding fluid.
The time-dependent FEA thermal lens signals in Figure 2-4 were calculated
assuming dn/dT = 1.0 for both liquid and glasses. The FEA models of the photothermal
heating of the glass samples showed significant heat transfer between the glass and air.
Boundary conditions normally assumed in thermal lenses are that this heat transfer is
negligible.
a
1.6E+06
Inverse focal length (1/m)
1.4E+06
1.2E+06
1.0E+06
8.0E+05
6.0E+05
4.0E+05
2.0E+05
0.0E+00
0.000
0.100
0.200
0.300
0.400
Tim e (s)
Figure 2-4a. FEA modeling of photothermal lens of FeCp2 solution.
31
b
6.0E+01
Inversefocal length(1/m)
5.0E+01
4.0E+01
3.0E+01
2.0E+01
1.0E+01
0.0E+00
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
Time (s)
Figure 2-4b. FEA modeling of photothermal lens of 3.05 mm thick glass filter.
c
3.0E-03
Inversefocal length(1/m)
2.5E-03
2.0E-03
1.5E-03
1.0E-03
5.0E-04
0.0E+00
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
Time (s)
Figure 2-4c. FEA modeling of photothermal lens of air around the glass filter.
Figures 2-4b and 2-4c show that the magnitude of the signal of glass is far greater than
that of the signal of the air (thermal coupling fluid). The photothermal signal in air is due
32
to the diffused heat from the glass. If we compare the magnitude of the photothermal
signal in glass (Figure 4b) and in the surrounding air (Figure 2-4c), the signal in glass is
about four orders greater than that in air. This is because the temperature change with
pulsed laser excitation is much faster than the heat diffusion to the surroundings.
Figures 2-4b and 2-4c show that the signal reaches maximum in glass very quickly
but in air it takes some time to reach the highest value. This is due to the fact that the
diffusion of heat from glass to air is a relatively slow process in comparison to the
excitation process.
7.0E+05
Inverse Focal Length (1/m)
6.0E+05
5.0E+05
4.0E+05
3.0E+05
2.0E+05
1.0E+05
0.0E+00
0.0000 0.0001 0.0001 0.0002 0.0002 0.0003 0.0003 0.0004 0.0004 0.0005 0.0005
Time (s)
Figure 2-5. Plot of semi-infinite cylinder (dashed line) and FEA modeling of glass with
(solid line) and without (dotted line) air boundary.
33
Figure 2-5, the FEA modeling of glass with (no transfer of heat from glass to air)
and without (transfer of heat from glass to air) boundary conditions clearly shows that
there is no heat transfer from glass to the air. Of interest here is, first, as expected, that
there is very little difference between the semi-infinite cylinder approximation and the
FEA model for the case where no heat transfer occurs between glass and air. Second,
including glass-air heat coupling results in an initial rapid decrease in inverse focal length
followed by decay closely resembling the semi-infinite cylinder case. The rapid decay is
due to heat loss to the air at the surfaces, which decreases the length of the photothermal
lens.
The signal magnitudes (1/f) at zero time, by FEA modeling and the analytical
equation of the photothermal lens, are within two percent of each other. This is quite
natural because FEA modeling depends on the mesh number, size of elements, and the
types of solver used for the process.
We propose a simple method for comparing and correcting the measured
photothermal lens signals in order to compare the different types of samples. FEA
analysis is thought to be more accurate than the approximate solutions. In particular, the
infinite cylinder approximation is clearly wrong for the glass samples. The photothermal
signal is modeled for the glass by FEA modeling with the boundary condition (no transfer
of heat from glass to air) and without the boundary condition (transfer of heat from glass
to air). The ratio of the modeled signals should be equal to the ratio of the experimental
signals without transfer of heat to transfer of heat. Similar measurements can be done for
34
the time constants to use the semi-infinite cylinder approximate model confidently for
photothermal lens experiments.
S bm
S
 be
S wm S we
(11)
Here, Sbm and Swm are the photothermal signals of FEA modeling with and without heat
coupling and Sbe and Swe are experimental signals with and without heat coupling with the
surroundings. A similar relation is true for time constants below
tc ,(bm)
tc,( wm)

tc ,(be)
tc ,( we)
The values of dn/dT for the glasses were estimated by comparing data obtained from the
thermal lens signals for the glasses and the well-established photothermal properties of
ethanol. At steady-state conditions, Eq. 5 indicates that the thermal lens signal is
proportional to the integral of the second derivative of the radial temperature profile. The
integration term from Eq. 5 is obtained directly from the finite element analysis modeling
solution.
Comparing Eq. 8 for the glass and the standard we have the following equation
(12)
35
 dn 
 dn  S g  g  s C p , g

 

 dT  g  dT  s S s  s g C p ,s
(13)
The subscripts ‘‘g’’ and ‘‘s’’ refer to glass and standard, respectively. The dn/dT for the
glasses was estimated by using Eq. 13. Corrected values for material were obtained by
using Eqs.11, 12, and 13. The experimental thermal diffusion constant calculated for
glass using the proposed equation gives excellent agreement with the theoretical value
obtained from the given values of density, specific heat capacity, and the corrected time
constant obtained from experiment and modeling.
Table I, shows the calculated dn/dT and the temperature coefficient of electronic
polarizability, σ (K-1), and is given by
 
1 d
 dT
for the glasses. Given β and n, from Eq. 10, the σ was calculated using estimated dn/dT.
The accuracy of the result was due to the calibration with the liquid samples and the fact
that the transmission of the liquid sample was measured with a conventional
spectrophotometer. The values of the polarization coefficient are in agreement with
Prod’homme’s report on silica glasses with low thermal expansion. This accounts for the
fact that electronic polarization is the dominant factor contributing to refractive index
variation. An increase in electronic polarization causes the refractive index to increase in
silica glasses.
(14)
36
Table 2-1. Thermo-optical constants of optical glasses.
Properties
Ethanol Corning CS3-73 (3.05 mm) Corning 3389 (1.45 mm)
5
-1
dn/dT (×10 K )
5
-1
5
-1
β (×10 K )
n
σ (×10 K )
α (m-1)
κ (W m-1 K-1)
-40
1.53
5.72
99
2.67
2.67
1.361
1.506
1.506
2.1
0.167
5.2
3.8
1.38
12.2
1.7
1.38
The thermo-optical coefficient, dn/dT, is negative for liquid samples, and for solids
and glasses it may be positive or negative due to many counteracting effects.30,31 Some
researchers have explained the factors contributing positive refractive index gradient in
solids.18,32 The nonlinear absorption coefficient and the index of refraction change in
semiconductor-doped glass are due to the photo-darkening effect,19,20 and it is attributed
to a photochemical process in the semiconductor microcrystallite.33 Trends in the
nonlinear properties of semiconductor-doped glass may depend on the band-gap
wavelength and doped particle sizes.34–37 The nonlinear refractive index change is
expected to be positive in the semiconductor-doped glass at excitation wavelengths below
the band-gap wavelength and negative at wavelengths above the band-gap wavelength.
The signal for glass filters that are doped with other materials rather than the
semiconductor microcrystals has been shown to have positive dn/dT and shows that the
positive dn/dT cannot only be a result of the semiconductor microcrystals.18, 38 The
excitation beam wavelength falls below the CdSxSe1-x band-gap wavelength, which
produces a positive refractive index change due to increased carrier density.
37
Thermal lens experiments and FEA modeling are presented to investigate the
temperature profile, change in refractive index of CdSxSe1-x doped glass, and to
determine the thermo-optical coefficient, dn/dT.
We previously reported the photothermal lens signal for similar glasses using
continuous laser-excited photothermal spectrometry.38 One of the problems realized at
that time was that the heat transfer from a glass plate excited with a laser is far from the
ideal situation described by the semi-infinite cylinder approximate models that are most
often used to describe photothermal lens experiments. Here we have resolved the
problem using: (1) pulsed laser excitation where the signal decays is faster than the heat
diffusion to the surroundings, and (2) FEA modeling to take account of the likely
deviation from the semi-infinite cylinder approximate models. The measured dn/dT and
the temperature coefficient of electronic polarizability values of glasses are very
reasonable compared to our previous labs reports38 and are much nearer to the report by
Prod’homme on silica glass.
The anomalous behavior of the temperature-dependant refractive index changes in
glasses has previously been explained in terms of optical nonlinearities and stress
birefringence. However, this is still is not a well-known phenomenon. Finite elemental
analysis modeling may help in understanding the dynamics of the temperature
distribution with other experimental parameters. Research has been done on surfacedeformation phenomena due to the excitation source in glasses.39, 40 FEA modeling and
the study of the transmission profile of the probe beam along with the surfacedeformation phenomenon due to excitation may help to better explain the secret of
anomalous behavior of the doped glasses.
38
CONCLUSION
Finite element analysis modeling can be used to better understand the pulsed laser
excited photothermal spectroscopy signals by correctly accounting for the heat transfer of
the sample to the surroundings. FEA modeling is particularly useful to account for
surface heat transfer that will occur in glasses and other samples where the absorbing
material is in direct contact with the coupling fluid surroundings. The cylindrical
approximation is not valid in these cases. Anomalous dn/dT behavior of these glass filters
is thought to be due to counteracting effects: optical nonlinearity, stress induced
birefringence, and the structural network in the glass structure. Further investigation,
using FEA modeling to take into account physical effects such as surface deformation
due to thermal expansion, will be required to understand the thermo-optical phenomenon.
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42
CHAPTER 3
DETECTION OF FAST AND SLOW SIGNAL DECAY COMPONENT IN GLASS BY
PHOTOTHERMAL EXPERIMENT USING FAST RESPONSE DETECTOR
ABSTRACT
Finite element analysis modeling of the photothermal lens signal produced with
Gaussian pulsed laser excitation of colored silica glass plates surrounded by air is
presented. Models show fast and slow time-dependent signal decay components present
in photothermal lens produced within the glass. Experiments are carried out to verify the
model prediction. Transmission and reflection photothermal lens experiments are carried
out to detect the event using fast response detector. The fast and slow decay signal
components are experimentally verified and interpreted as being due to the fast transfer
of heat in axial direction and slow transfer of heat in radial direction relative to the
excitation laser beam. The time constant of slow and fast components is evaluated to
present the correct prediction of material properties with the help of the finite element
analysis model.
INTRODUCTION
Thermal lens spectrometry (TLS) is a photothermal technique, which has been
widely used for the determination of very low optical absorption coefficients of different
materials with high sensitivity and versatility. 1–4 Since its discovery by Gorden et al.5 in
1964 the method has been used for the intracavity measurements of absorption of nearly
transparent materials. 1, 2 Knowledge of the thermo-optical properties is essential for
43
evaluating the figures of merit of optical glasses which can readily be used to
determine the working glass conditions6,7 such as thermal shock, thermal stress
resistance,8 and thermal lens effect.6-9
CdSxSe1-x-doped glass was used as a sample in this work. In the past few years,
semiconductor doped glasses and other analogous materials with semiconductor quantum
dots have been investigated intensively and many physical phenomenon have been
found.10-16 In particular CdSxSe1-x-doped glasses have received considerable attention as a
promising nonlinear material with a large optical nonlinearity17and fast optical
response.18 Theoretical models of TLS effects in fluids are well established. But accurate
theoretical descriptions of thermal diffusion in solids excited by laser beams is still an
active area of research although a number of approximations are commonly used to
obtain tractable analytical results.19,20 The use of thermal lens techniques for analysis
glasses has been reported by some authors with the help of finite element analysis. 21, 22
The limit of detection for thermal lens methods are related to how realistically the
experimental description can be theoretically modeled.19, 20 Generally, simple and
applicable theoretical model are obtained by introducing modeling approximations that
lead to analytical solutions. In some cases, these approximations can be accounted for by
using appropriate experimental setups. However, it is not always feasible. In most
practical situations, for example, heat is transferred from sample to the surroundings
along to the axial dimension. This is especially true when the heated sample is direct in
contact with air or another coupling fluid. Recently, it has been shown that sample/air
heat coupling in the thermal lens experiment could significantly contribute to the thermal
lens signal.21, 23 In fact; the fluid thermal coupling is always treated as a perturbation to
44
the thermal lens signal. This perturbation could become stronger as the sample
thickness is reduced and /or depending on the thermo-optical properties of the sample/
fluid system.
Conventional TLS models treat the sample as an infinite medium, 5,24 both in radial
and axial directions (in cylindrical coordinates), which has the weakness that the
temperature rise in the sample never reaches steady state.5 Wu and Dovichi25 derived a
cw laser-induced single-beam TLS model for their steady-state thin-film measurement
which takes the axial heat loss into account. Accordingly, when the sample is thin the
axial heat flow cannot be neglected and the axial-infinite treatment is no longer valid. In
their work the sample in the radial direction is still assumed to be infinite. However there
are two ways to remove the heat energy from the illuminated sample: axial and radial
heat flows into the surrounding air which depends on the sample boundary conditions.
The theoretical model for the thermal lens signal is difficult to deduce analytically
when heat coupling with the surrounding in taken into account using more realistic
boundary conditions. Such models can be obtained with approximations but the lack of a
more complete theory makes validation difficult. FEA methods provide numerical
solutions to the heat transfer equations with the realistic boundary conditions imposed by
the experimental geometry. FEA software has been recently applied to describe complex
heat coupling conditions on photothermal techniques. 21, 23 Numerical and approximated
solutions have been compared to FEA modeling. FEA has been shown to be a powerful
tool to model continuous and pulsed-laser photothermal methods.
This work presents FEA methods for modeling of the pulsed laser excited
photothermal lens signal of glass samples surrounded by air. The model accounts for the
45
coupling of heat both within the sample and out to the surroundings. While studying
the FEA model of pulsed-laser excited thermal lens signal of colored glass filters
surrounded by air, a kind of irregular decay of signal was predicted. On a close scrutiny it
was realized that it could be due to fast and slow heat transfer components. Transmission
and reflection photothermal lens experiments using fast response detector were carried
out to verify and detect these events. These fast and slow decay signals can be explained
in terms of axial and transverse heat diffusion from the sample.
THEORY
Thermal lens spectrometric method is based on the periodic formation of a lenslike element in a medium that absorbs laser radiation with the TEM00 mode. The focal
distance of the resulting thermal lens depends on the absorbance of the sample. The
thermal lens signal depends on the temperature gradient of the refractive index and the
thermal conductivity coefficient of the medium. The energy absorbed by the sample for a
collimated, short-pulsed Gaussian profile laser propagating on the z-axis is26
q H ( r , z , t )   e  z
2Q  ( 2 r 2 / w 2 )
e
 (t )
w2
(1)
The z-axis origin is taken to be at the entrance interface of the sample. In this
equation, α is the sample exponential absorption coefficient. The first part of the energy
source equation is z-axis energy deposition due to sample absorption of the exponential
attenuated laser light as it traverses the sample. The second part of Eq. 1 accounts for the
radial distribution of energy produced by the Gaussian laser profile. Q (J) is laser energy,
46
YH is a heat yield parameter accounting for energy loss dues to luminescence during
sample excitation, and w (m) is the excitation laser beam electric field width. It is
assumed that the laser beam waist does not change over the length of the sample. The last
term in this equation is the time dependence, in this case a delta function, δ(t), which is
zero unless t = 0.
The spatial heat distribution in the sample in the absence of radiation losses is
defined by the three-dimensional differential equation of thermal diffusion.26
q (r , z, t )

T (r , z , t )  DT  2T (r , z, t )  H
C p
t
(2)
Here r and z are the radial and linear cylindrical coordinates, CP is the heat capacity, ρ is
the density of the medium, δ T (r, z, t) is the change in the sample temperature with
respect to the initial level, z is the coordinate along the axis coinciding with the beam
propagation direction, t is the time, and DT is the thermal diffusion constant.
The infinite cylinder approximation for excitation pulse duration much shorter than
the thermal relaxation time, and assuming rapid excited-state relaxation, the analytical
solution to the temperature change is given by Eq. 3 for t ≥0:
T (r , z, t )  T ( z , t ).T (r , t )
47
2YH Q
1
e
T (r , z, t )  T ( z , t )
2
C p w (1  2t / tc )


2r 2


 w 2 1 2 t / t  
c 

(3)
tc (s) is the characteristic thermal diffusion time constant defined by
tc 
w2
4 DT
(4)
The strength of the photothermal lens element is found from the second radial
derivative, evaluated on-axis. Integration over path length results in the inverse lens
strength.
1
d2
 ds 

  T ( z , t ) 2 T (r , t ) ds
f (t )  dT  path
dr
r 0
(5)
where f (t) is focal length and ds/dT is the thermo-optical coefficient allowing for
pathlength expansion also known as the thermal-optical coefficient.
If the sample is thick and surface is insulated than there is no diffusion of heat
through the surface and there no is axial temperature change. In most practical situations
heat is transferred from the sample to the surroundings along the axial dimension. This is
especially true when the heated thin sample is in contact with the air or another coupling
fluid. In that case δT(z, t) also contribute to the total signal. This surface diffusion along
axial direction is responsible for initial fast decay of signal.
48
The change of divergence of a laser beam passing through the sample, due to the
thermal lens effect, can be measured simply by placing a pinhole over the optical detector
in the far field in the center of the diverged beam. The relative change at the detector
signal is proportional to the inverse of the focal length. The photothermal lens signal is
related to the inverse focal length. For pulsed laser excitation, the experimental
photothermal signal, S(t), is defined by26
S (t ) 
 ()   (t )
 (t )
(6)
In Eq. 6, Φ is the irradiance of the probe laser passing through a pinhole aperture
placed far away from the sample and the probe laser is focused a distance z’ in front of
the sample. The signal is proportional to the inverse focal length of the lens produced in
the sample due to laser heating; S(t) = 2z’/f(t). This is an approximation and better signal
predictions may be made with diffraction theory.
MATERIALS AND METHODS
Experimental apparatus. A conventional two-color photothermal lens apparatus,
with probe laser 632.6 nm HeNe, and excitation source XeCl excimer dye pulsed laser
operating at 490 nm was used. A detailed description of experiment and apparatus used
can be found elsewhere.21 However; we have used fast response photodiode as a detector.
The Figure 3-1 illustrates the apparatus set up for photothermal lens experiment.
49
Figure 3-1. A schematic drawing of a photothermal lens experimental setup.
Sample. Standard Corning colored glass optical filters doped with cadmium
sulfoselenide microcrystals of about 10 nm are investigated as an absorbing solid
material. 5 cm x 5 cm Corning 3389 (CS3-73, 3.05 mm thickness) optical filters was used
for glass samples. The sample is positioned at the focus of the excitation beam for
maximum temperature gradient
Modeling. Photothermal signals are modeled for heat transfer from glass to air
after laser excitation and the FEA model results are subsequently compared to the
experimental signals. FEA is a tool for numerical solutions to complex differential
equations. Comsol Multiphysics 3.5a analysis is carried out on a Compaq Presario
SR1330X, AMD Athlon XP 3200 processor using MS Windows XP.
50
The Comsol Multiphysics software (v.3.5a) in conduction and convection mode solves
the heat diffusion equation given as
C p

T (r , z, t )  k 2T (r , z, t )  qH (r , z, t )  C p u  T (r , z, t )
t
(7)
where u is the flow velocity. Note that Eqs. (2) and (7) differ only by the second term on
the right side of the equation. This term can account for convection or mass flow heat
transfer, which is not important for the solid sample investigated in this work.
A detailed description of FEA modeling has been described elsewhere.21Briefly; it
consists of drawing the sample geometry and specifying material boundary conditions,
heat sources, and sinks. The problems are then solved with rough finite element
definition and further refinement of elements and domain are made. The element mesh is
refined until model results become independent of mesh size. Finally, the temperature
profile can be obtained either at a single time, over a time series, or at steady state, for the
sample and air domain. The path integral of the second radial derivative was found by
using the Comsol integration-coupling variable to integrate the second derivative function
of the temperature change.
For convenience, glass heating was modeled with 2D axial symmetry with the
origin at the center that the z-axis runs along the pathlength of the excitation and probe
beams. We have tested this and found that the results are equivalent to those obtained in
the three dimensional Cartesian model. However, the cylindrical model is smaller and
51
quicker to solve. The model solid-absorbing media taken were cylindrical plate of 5
mm diameter and 0.5 to 3.05 mm of different thickness.
RESULTS AND DISCUSSION
Figure 3-2 is temperature profile along the z direction (air-glass-air) using the FEA
modeling, and clearly shows that there is heat transfer from glass to air.
Figure 3-2. FEA modeling of glass surrounded by air (2D axial symmetry).
52
Figure 3-3 shows the FEA modeling of photothermal signal of 3.05 mm thick
glass surrounded by air. The plot is inverse focal length versus time for 40-100 µm beam
width pulse-laser excitation. Clearly, Figure 3-3 illustrates that the signal increases with
decreasing beam size. The interesting thing here is that the glass-air heat coupling is
probably responsible for the faster initial decay of the time-resolved photothermal signal.
The slower decay is due to radial heat diffusion. The faster signal decay becomes
prominent when the beam size decreases. The faster signal decay is apparently due to the
axial heat transfer. There is no fast signal decay for the thick samples indicating less
contribution from axial heat transfer to the total cooling.
1.80E+06
1.60E+06
Inverse focal length (1/m)
100 um
1.40E+06
90 um
1.20E+06
80 um
70 um
1.00E+06
60 um
8.00E+05
50 um
6.00E+05
40 um
4.00E+05
2.00E+05
0.00E+00
0.0000
0.0001
0.0002
0.0003
0.0004
0.0005
Time (s)
Figure 3-3. FEA modeling of 3.05 mm glass surrounded by air using 40-100 µm
diameter pulsed-laser excitation.
53
The theoretical time-resolved photothermal lens signals can be used to evaluate
the tc and thus the thermal diffusion constant using DT = w2 / tc. The slower component of
the time-resolved photothermal signal is extrapolated and tc is calculated based on the
time to ½ maximum signal (t1/2). The glass DT obtained from this extrapolation method
gives a more realistic value than the one calculated from the Figure 3-4 which is the plot
of w2 versus tc.
Figure 3-5 indicates that the signal intensity goes down with thickness of glass and
the axial heat transfer is more severe. This is because of rapid diffusion of heat at the
surface.
0.0030
Time constant (s)
0.0025
0.0020
0.0015
0.0010
0.0005
0.0000
0
2000
4000
6000
8000
10000
Square of laser width (um2)
Figure 3-4. Variation of time constant (tc) with square of laser width (w2).
12000
54
3.00E+05
Inverse focal length (1/m)
2.50E+05
3.05 mm
2.00E+05
2.50 mm
2.00 mm
1.50E+05
1.50 mm
1.00 mm
0.50 mm
1.00E+05
5.00E+04
0.00E+00
0.0000
0.0001
0.0002
0.0003
0.0004
0.0005
Time (s)
Figure 3-5. FEA modeling of glasses of 0.50-3.05 millimeter thickness.
1.00E+05
9.00E+04
Inverse focal length
8.00E+04
7.00E+04
6.00E+04
glass signal
5.00E+04
air signal
4.00E+04
3.00E+04
2.00E+04
1.00E+04
0.00E+00
0.0000
0.0005
0.0010
0.0015
0.0020
Time (s)
Figure 3-6. FEA modeling of photothermal signal in 1 mm glass and air.
55
The heat transfer dynamics of glass surrounded by air is modeled and the
photothermal signal of glass and air are subsequently evaluated separately. The results
show that the air signal contribution to the total photothermal lens signal is significant.
This is shown in Figure 3-6. The signal in air, which is due to the transfer of heat from
the glass surface, presented in Figure 3-7, does not change with the size of glass.
Transmission and reflection photothermal lens experiments were carried out by
using fast response photodiode detector on 3.05 mm glass. The experimental schematic is
shown in Figure 3-8. The fast response signal has been detected. Transmission and
reflection signal look similar which indicates that they are due to the axial heat transfer.
1.20E+04
Inverse focal length (1/m)
1.00E+04
8.00E+03
2.50 mm
6.00E+03
2.00 mm
1.50 mm
1.00 mm
4.00E+03
0.50 mm
2.00E+03
0.00E+00
0.0000
0.0001
0.0002
0.0003
0.0004
-2.00E+03
Time (s)
Figure 3-7. FEA modeling of photothermal signal in air.
0.0005
56
0.05
0
0
0.00005
0.0001
0.00015
0.0002
0.00025
Signal
-0.05
-0.1
Transmission
Reflection
-0.15
-0.2
-0.25
Time (s)
Figure 3-8. Experimental photothermal signals (transmission and reflection).
CONCLUSION
Finite element analysis modeling has been used to describe the pulsed laser
excited photothermal lens signal by considering both the heat transfer from sample to air
and the thermal lens generated in the air surroundings.
The results show that the air signal contribution to the total photothermal lens
signal is significant. It also shows that there are slow and fast signal decay component in
glass. Transmission and Reflection Photothermal lens experiments have been carried out
using fast response detector and the fast signal decay is detected. The fast decay of signal
is interpreted to be due to the fast transfer of heat in axial direction, the direction of the
excitation laser beam.
57
When the heated sample is direct in contact with air or another coupling fluid,
heat is transferred from sample to the surroundings along to the axial dimension. The
fluid thermal coupling is treated as a perturbation to the thermal lens signal. Here this
perturbation appeared in the form of initial faster signal decay. It has been shown that
FEA modeling could be used to correct the perturbation to the thermal lens signal for
material parameter determination.
These studies open up the possibility of application of FEA modeling and the
pulsed excited thermal lens method for accurate prediction of the heat transfer to the
coupling fluid and subsequently to study the gas surrounding the samples by using a
known material solid sample.
REFERENCES
1. Leite R.C.C., Moore R.S., and Whinnery J.R., Appl. Phys. Lett, 5, 141 (1964).
2. Solominini D., J. Appl. Phys., 37, 3314 (1965).
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4. Marcano A.O., Loper C., and Melikechi N., J. Opt. Soc. Am. B, 19, 119 (2002).
5. J. P. Gordon, R. C. C. Leite, R. S. Morre, S. p. S. Porto, and J. R. Whinnery, J.
Appl. Phys. 36, 3 ( 1965).
6. S. A. Payne, C. D. Marshall, A. Bayramian, G. D. Wilke, and J. S. Hayden, Appl.
Phys. B 61, 257 (1995).
7. N. Neuroth, Opt. Eng. 26, 96 (1987).
8. P. Greason, J. Detrio, B. Bendow, and D. J. Martin, Mater. Sci. Forum 6, 607
(1985).
58
9. M. Sparks, J. Appl. Phys. 42, 5029 (1970).
10. N.F.Borrelli, D.W. Hall, H.J. Holland, and D.W. Smith, J. Appl. Phys. 61, 5399
(1987).
11. M.G. Bawendi, P. J. Carroll, W. L. Wilson, and L. E. Brus, J. Chem. Phys. 96,
946 (1992)
12. T. Miyoshi, Jpn. J. Appl. Phys. 31, 375 (1992).
13. P. Roussignol, D. Richard, C. Flytzanis, and N. Neuroth, Phys. Rav. Lett. 62, 312
(1989).
14. M. C. Klein, F. Hache, D. Richard, and C. Flytzanis, Phys, Rev. B 42, 11 123
(1990).
15. J. J. Shiang. S. H. Risbud, and A. P. Alivisatos, J. Chem. Phys. 98, 8432 (1993).
16. L. E. Brus, J. Chem. Phys. 80, 4403 (1984).
17. R. K. Jain and R. C. Lind, J. Opt. Soc. Am. 73, 647 (1983).
18. J. Yumoto, S. Fukushima, and K. Kubodera, Opt. Lett. 12, 832 (1987).
19. A. B. Chartier and S. E. Bialkowski, Opt. Eng. 36, 303 (1997).
20. J. R. Whinnery, Acc. Chem. Res. 7, 225 (1974).
21. P. R. Joshi, O. O. Dada, and S. E. Bialkowski, Appl. Spectrosc., 63 815 (2009).
22. O. O. Dada, M. R. Jorgensen, and S. E. Bialkowski, Appl. Spectrosc. 61, 1373
(2007).
23. O. O. Dada, and S. E. Bialkowski, Appl. Spectrosc. 62, 1326 (2008).
24. J. Shen, R. D. Lowe, and R, D. Snook, Chem. Phys. 165, 385 (1992).
25. S. Wu and N. J. Dovichi, J. Appl. Phys. 67, 1170 (1990).
59
26. S. E. Bialkowski, Photothermal Spectroscopy Methods for Chemical Analysis
(John Wiley and Sons, New York, 1996).
60
CHAPTER 4
MEASUREMENT OF SURFACE DEFORMATION OF GLASS BY PULSED-LASER
EXCITED PHOTOTHERMAL REFLECTION LENS EFFECT
ABSTRACT
A time resolved reflection photothermal lens method for the measurement of
thermo-mechanical properties of glasses is presented. Finite elemental analysis software
with a thermal-structural module is used to model the thermal lens effect by simulating
the coupling of heat both within the sample and out to the surroundings. It is shown that
the thermal lens produced in the coupling fluid can have significant influence on the total
lens strength in air but can be greater for the coupling fluid with larger thermo-optical
coefficient. An experimentally monitored thermoelastic surface displacement result is
found to be in agreement with the predictions of the finite element analysis model.
INTRODUCTION
Photothermal methods have been used to determine thermal and optical properties
of materials since 1981 when photothermal beam deflection or “mirage effect” was first
described.1 There has since steady improvements to both theory and experiment. Several
pulsed laser excited methods for measuring optical absorptance and thermal expansion
properties of transparent and opaque solids have been described.2-4 Applications include
semiconductors and nanomaterials.5
61
In order to measure optical and thermo-physical properties of solid in
microscopic scale, photothermal spectroscopy is considered to be the one of the most
widely applicable techniques because of its noninvasive in nature. Noncontact optical
methods are preferred over traditional methods because of the simplicity of sample
preparation and the absence of thermal contact problems.6 A microscopic capability also
provides the advantage of using a small sample and the potential for scanning or mapping
of the properties on a surface.
A tightly focused pulsed laser beam is most often used to excite the solid sample in
photothermal spectroscopy. In addition to photothermal effect, one of the most ordinary
observed effects is the surface displacement or deformation due to optical excitation.
Absorbed energy converts to heat; resulting in thermoelastic expansion (or contraction)
of the heated sample thereby producing a curved mirror-like change in the sample
structure. This structure acts as a lens altering the divergence of the probe laser. The
deformation behaves as a concave or convex mirror 4,7-10 depending on the optical and
thermomechanical properties of the sample, such as thermal expansion and/or electronic
polarizability coefficients. The displacement is measured by the divergence of the
surface-reflected probe laser beam.
The term photothermal reflection lens was first used by Bialkowski at the Gordon
Research Conference on Photoacoustic and Photothermal Phenomena (Trieste, IT 2005)
for thermoelastic surface displacement produced optical component.11 But it is now
called photothermal mirror by some researcher.12-15
There are several ways to monitor the effect of sample heating. Among them are
photothermal radiometry, photothermal deflection or “mirage effect,” and measuring
62
divergence of surface-reflected laser beam. The latter may be having advantages
relative to the measurement of beam angle changes used in the photothermal
displacement apparatus. The most important advantage is the relative insensitivity to the
position of excitation and probe laser beams on the surface.
Modulated irradiation produces a frequency-domain signal. Changing the
modulation frequency results in signal magnitude and phase angle changes relative to the
excitation. Thermo-elastic properties of the sample are derived by the frequencydependent amplitude and phase angle changes. On the other hand continuous and pulsed
irradiation heating results in time-domain signals. The thermoelastic properties are
extracted though analysis of the time-dependent signals. Analysis of the signal, either
frequency- or time-dependent, yields the thermal diffusivity and expansion coefficient.
Analysis of the transient time-dependent surface displacement and/or temperature change
signals allows the measurement of thermal diffusion coefficient.
Bennis et al. and Li solved for the thermoelastic displacement following pulsed
irradiation for the case where no heat is conducted to the fluid at the surface and for the
case where there is a large surface absorption.4-5 The excitation beam is incident along
the z direction and is absorbed at the surface of the sample. The back-reflected
photothermal mirror observes the resulting deformation. Photothermal displacement will
produce a lens-like mirror with a time-dependent inverse focal length. And solving the
Navier-Stokes equation with Duhamel’s approximation leads to the term required
calculating the inverse focal length.
Although the physical basis is given in Bennis et al. and Li’s approximations,
better analysis can be done with more realistic models. Two shortcomings of models
63
described above are 1) that heat transfer to the coupling fluid (the gas or liquid at the
sample surface) has negligible effect on the time-dependent temperature and
displacement and 2) that refractive index changes in the coupling fluid does not affect the
photothermal measurements. To our knowledge, solutions to the time-dependent thermal
and thermoelastic where heat is coupled to the surroundings are not known for pulsed
laser irradiation.
The cylindrical-symmetric finite element analysis FEA simulation for Corning
glass in air shows that there is significant amount of heat coupling to air.16 The surface
temperature is less than that at the interior of the glass. In the absence of the coupling
fluid the surface temperature would be at the maximum. Thus models that do not account
for heat transfer to the surroundings are in error and the time-dependent surface
temperature measurements would not give an accurate estimate for thermal diffusivity
because the surface cools faster than these models predict. Another point is that heat does
not diffuse to the back surface of a thick glass on the short small timescales of the
transient surface displacement.
The aim of this work is to improve measurements of optical and thermal properties
of solids by using a pulsed excitation and utilizing photothermal reflection lens effect to
monitor thermoelastic surface displacement and by using more accurate physical models
that account for surface heat transfer to the surrounding gas or fluid.
THEORY
In pulsed-laser excited photothermal measurements, a short pulse of light excites
the surface of the sample. Energy absorbed and not subsequently emitted as luminescence
64
results in surface heating. The initial temperature change is δT=HαYH/ρCP. Here H is
radiant exposure, α is the optical absorption coefficient, which for surface absorption is 1R where R is reflectivity and YH is the heat yield or absorptance. Heat subsequently
dissipates through both the sample and the gas or fluid in contact with the sample surface
changing both the surface temperature and that of the contact fluid.
Bennis et al. and Li solved for the thermoelastic displacement following pulsed
irradiation.4-5The excitation beam is incident along the z direction and is absorbed at the
surface of the sample. The back-reflected photothermal mirror is used to observe the
resulting deformation. Photothermal displacement will produce a lens-like mirror with a
time-dependent inverse focal length given by
1
 2u z (t )
 2
f (t )
r 2
r0
(1)
where uz(t) is the z-component of the thermoelastic displacement vector u(r,z,t). The
thermoelastic deformation vector is found from the solution of
 2u(r, z, t )
(   )(  u(r, z, t )   u(r, z, t )   0
  t (3  2 )T (r, z, t )
t 2
2
Here, T(r, z, t) is the temperature change, λ and μ are Lamé coefficients describing the
time-dependent elastic deformation, ρ0 is density, and αt is the coefficient of thermal
(2)
65
expansion. For times longer than the acoustic relaxation time, Duhamel’s
approximation leads to static equation
(  u(r , z , t )  (1  2 ) 2 u( r , z , t )   t (1   )T ( r , z , t )
(3)
ν is Poisson’s ratio.
The temperature follows the thermal diffusion equation solved with appropriate boundary
conditions
T
U (r , z, t )
(r , z , t )  DT  2T (r , z , t ) 
t
 0C p
(4)
U(r,z,t) is the energy density, and DT = kT/ρ0CP is the thermal diffusivity where kT is
thermal conductivity
Initial surface heating. For a Gaussian excitation laser with a radiant exposure, the
distribution is
H (r , t ) 
2Q ( 2 r 2 / w2 )
e
T
w 2
(5)
The energy density for a short-pulsed excitation laser is
U (r , z, t ) 
2QYH ( 2 r 2 / w2 ) ( z )
e
e
 (t )
w 2
(6)
66
α is the optical absorption coefficient, Q is the pulsed laser energy, w is the electric
field beam waist radius, and YH is the heat yield. The heat yield may be related to
reflectivity through YH =1- R where R is reflectivity.
Time-dependent temperature and thermoelastic displacement. The thermal
diffusion and thermoelastic displacement equations have been solved for the case where
no heat is conducted to the fluid at the surface and for the case where there is a large
surface absorption.4-5The temperature change can be expressed by

T (r , z, t )  C1  J 0 (r )e
(
2 w
8
)
0
where
2
2


  N k ( , k ) cos( k z)e{ DT ( k )(t  )} d
 k 0

N k ( ,  k )  [1  e {  DT ( 
M k (  , k ) 
and
2
 k2 ) }
(7)
]M k (  ,  k )
2  2 [1  (  1) k e (   L ) ]
( 2   k2 )(  2   k2 )
ηk = πk/L where L is the sample thickness, C1= (QYH/2πkTL τ) and τ is the laser pulse
duration. Similarly, solving the Navier-Stokes equation with Duhamel’s approximation
leads to the term required to calculate the inverse focal length
 2 u z (t )
r 2

r 0
 2 C 1 t (1   )  e
0
(
2 w 2
8
)
2
2


X   Ak (  , k ) e {  DT (   k )( t  )}  d 
 0

(8)

Ak ( , k )  2 1  e
where
B k (  , k ) 
and
{  DT ( 2  k2 ) }
4
(2   k2 ) 2
B ( , )
k
67
k
 cosh(L)  (1) k 


k
 sinh(L)  (1) L 
This model shows that the magnitude of the reflection lens signal will be
proportional to αt(1+ν) while the time-dependent behavior is related to DT. There are
facile forms of the above results for high absorption coefficient, short pulse duration, and
for samples with large L.
Heat transfer with coupling fluid. Solutions to the time-dependent thermal and
thermoelastic where heat is coupled to the surroundings are not known for pulsed laser
irradiation. The simplest is for one-dimensional thermal diffusion along the z-direction
T ( z, t ) 
2
Q Y H
1 / 2 (  z / 4 DT , f t )
(

t
)
e
 s C p , s DT1 /, s2   f C p , f DT1 /, 2f
Subscripts s and f refer to solid and fluid respectively. Measurements of the timedependent surface temperature (z=0) are used to determine the thermal relaxation time,
which in turn is related to the thermal diffusion and heat capacity of the solid and fluid.
When these properties are known for the contact fluid and the parameter ρS CP,S DT,S1/2 =
(ρS CP,S kT,S)1/2 is derived from the time-dependent temperature decay. Duhamel’s NavierStokes approximation is used to determine displacement if the temperature distribution is
known.
(9)
68
The temperature change and thermoelastic displacement can be calculated using
finite element analysis (FEA). FEA modeling using Comsol Multiphysics software in
conduction and convection mode solves the heat diffusion equation given above. The
thermoelastic equation is coupled to the temperature change. Hook’s law models the
time-dependent expansion with parameters approximating the Lamé coefficients
describing the time-dependent elastic deformation.
Photothermal reflection lens. In photothermal reflection lens or mirror apparatus,
the excitation and probe laser beams strike the sample in collinear arrangement. There is
no displacement between the two beams at the sample. Practically speaking, beam
alignment is easier because the signal is optimized when the two beams are collinear.
Thermoelastic surface displacement acts as an optical element modeled as a defocusing
mirror. Thus the spot size of the reflected probe laser changes with surface deformation.
The spot size can be calculated using ray transfer matrix methods based on the inverse
focal length of the mirror.6
The photothermal lens element is analyzed by directing the probe laser beam at the
sample. The reflected beam is detected past a pinhole aperture through a beam splitter set
to direct the reflected beam to the pinhole and detector. Treating the displacement as a
lens, the power of the probe laser beam passing through an aperture place in far field is
the photothermal lens signal. The signal is defined as S(t) = -2z’/f(t) where z’ is the
sample-detector distance.
Effects of coupling fluid refractive index change. In general, the inverse focal
length consists of two terms: 1/f(t) = 1/ff(t)+ 1/fs(t) where f and s refer to fluid and sample
reflection, respectively. The two terms are
69
2
1
 dn   T (r , z , t )
 2
ds

r 0
f f (t )
r 2
 dT  path
 2 u z (t )
1
2
f s (t )
r 2
(10)
(11)
r 0
Recent calculation have shown that the photothermal lens produced in the coupling fluid
can have significant (5%) influence on the total lens strength in air but can be greater for
the coupling fluid with larger thermo-optical coefficient, dn/dT.
MATERIALS AND METHODS
Experimental apparatus. A conventional two-color photothermal lens apparatus
with a probe laser 632.6 nm HeNe probe laser and a XeCl laser pumped dye laser
excitation source operating at 490 nm was used. Directing the collinear excitation and the
probe laser beam onto the sample surface performed the photothermal reflection lens
measurement. The beams were made collinear using a polarizing or dichroic
beamsplitter. In this way, one laser was transmitted and the other was reflected off the
front surface of the beamsplitter. Other mirrors were used to direct the beams onto the
beamsplitter and then onto the sample. A beamsplitter was placed along the beam
propagation axis. The back reflected probe laser was redirected toward a thermal lens
detection setup. The latter consists of an optical bandpass filter used to block the
excitation laser radiation, a pinhole aperture placed at the center of the probe laser beam.
70
Figure 4-1. A schematic drawing of a photothermal lens experimental setup.
The fast response photodiode signal was buffered with a transimpedence circuit,
amplified with a variable gain voltage amplifier and averaged over several repetitions of
the pulsed laser with a digital oscilloscope. The transient data were subsequently
transferred to a computer for data analysis. Figure 4-1 illustrates the schematic drawing
of a photothermal lens experimental set up.
Sample. Standard Corning, 5 cm x 5 cm, 3389 colored glass optical filters (CS3-
73, 3.05 mm thickness) of absorption coefficient 3.8m-1 were investigated as an
absorbing solid sample. The sample is positioned at the focus of the excitation beam for
maximum temperature gradient. Sample absorbance was recorded with a Cary 3E UVVisible spectrophotometer.
71
Modeling. Finite element analysis (FEA) software provides numerical solutions
to the heat transfer equations with the realistic boundary conditions imposed by the
experimental geometry. The model solid absorbing media was small square plates, 5 cm
x 5 cm x 0.305 cm, or 5 cm x 5 cm x 3.05 mm, and had the same dimensions of the glass
samples used in the experiments. Optical excitation was along the small dimension. For
convenience, glass heating was modeled with 2D axial symmetry with the origin at the
center that the z-axis runs along the path length of the excitation and probe beams.
Comsol Multiphysics in thermal structural interaction module (v 3.5a) analysis was
carried out on a Compaq Presario SR1330X, AMD Athlon XP 3200 processor using MS
Windows XP. The software in conduction and convection mode solves the heat diffusion
equation given as
C p

T (r , z, t )  k 2T (r , z, t )  q H (r , z, t )  C p u  T (r , z, t )
t
While the stress-strain analysis for axial symmetry uses Newton’s 2nd Law with
Rayleigh viscous damping.
 2u
 2    cu  F
t
2
 u
u
m 2 
 ku  f (t )
t
t
 r  rz  r   
 Kr  0


r
z
r
 rz  z  rz


 Kz  0
r
z
r
72
Finite element analysis modeling consists of drawing the sample geometry and
specifying material boundary conditions, heat sources, and sinks. The problems are then
solved with rough finite element definition and further refinement of elements and
domain are made. Finally, dT can be obtained either at a single time, over a time series,
or at steady state. The path integral of the second radial derivative was found by using the
Comsol integration-coupling variable to integrate the second derivative function of the
temperature change.
RESULTS AND DISCUSSION
Figures 4-2 and 4-3 are FEA modeling of thermal expansion of 3.05 millimeter
silica glass without and with air surrounding. Pulsed laser (4.4x10-4J) of 60 µm size was
used for modeling. The time is 0.01second after pulse irradiation. The extent of
temperature goes up from blue to red in figures. Two things are apparent from these
figures. First is the glass- air heat coupling resulting in heat loss from the sample to the
surroundings. Second, glass expansion is affected by coupling fluid because of the heat
transfer.
Figures 4-4 and 4-5 compares the temperature change in the glass with and without
the coupling fluids around it. Temperature change is slightly lower when the sample is
surrounded by air than insulated glass case. This is due to the heat diffusion to the air.
Similarly Figure 4-6 and 4-7 surface displacements along z direction, which is also a
direction of excitation laser, of glass without and with air environment. Again these
figures show that the thermal displacement of glass is affected by surrounding of glass
sample.
73
Figure 4-2. FEA model of thermal expansion of 3.05mm silica glass.
Figure 4-3. FEA model of thermal expansion of 3.05mm silica glass surrounded by air.
74
Figure 4-4. FEA model (plot of temperature change) of thermal expansion of 3.05mm
silica glass.
Figure 4-5. FEA model (plot of temperature change) of thermal expansion of 3.05mm
silica glass surrounded by air.
75
Figure 4-6. FEA modeling of (surface displacement along the z direction) silica glass.
Figure 4-7. FEA modeling of (surface displacement along the z direction) silica glass
surrounded by air.
76
Figure 4-8. FEA modeling of (second derivative of z displacement with respect to r vs
axial distance) silica glass.
Figure 4-9. FEA modeling of (second derivative of z displacement with respect to r vs
axial distance) silica glass surrounded by air.
77
4.00E+02
without air
with air
Relative inverse focal length (1/m)
3.50E+02
3.00E+02
2.50E+02
2.00E+02
1.50E+02
1.00E+02
5.00E+01
0.00E+00
0.000
0.002
0.004
0.006
0.008
0.010
0.012
Time (s)
Figure 4-10. FEA modeling of normalized relative inverse focal length versus time of
silica glass with and without air environment.
0.005
0.00000
-0.005
0.00005
0.00010
0.00015
R elative S ig n al
-0.015
-0.025
-0.035
-0.045
-0.055
Time (s)
Figure 4-11. Reflection photothermal lens signal.
0.00020
0.00025
78
Surface deformation is larger in insulating system of glass than in air
environment. With the help of FEA modeling second derivative of displacement with
respect to radial distance can be evaluated.
The second derivative of displacement with respect to radial distance of glass with
and without heat coupling surrounding can be compared from Figure 4-8 and 4-9.These
values are also different. FEA modeling of normalized relative inverse focal length
versus time of silica glass with and without air environment is presented in Figure 4-10.
And it shows that the magnitude of signal in glass with coupling fluid is larger than that
of without coupling fluid.
Figure 4-11 is an experimental reflection photothermal lens signal of Corning glass
filter 3389. It has positive optical-thermal coefficient and gives negative signal. The
inverse focal length measured from it is comparable that of the FEA modeling to estimate
the thermal physical properties of solid samples.
The models that do not account for heat transfer to the surroundings are apparently
in error. Using these models to determine optical and thermophysical properties of solids
by photothermal spectroscopic methods will result in inaccurate results. Taking account
of the contributing effect of coupling fluids should enhance the accuracy of the opticalthermo-physical properties of the solid sample in heat coupling environments when
analyzing experimental results obtained using photothermal reflection lens methods. The
inverse focal length of the system, consists of two terms, is sum of inverse focal length of
sample and coupling fluid. If we know the dn/dT of coupling fluid, inverse focal length of
it can be evaluated which can be used to correct the values. FEA modeling is helpful to
evaluate the inverse focal length.
79
CONCLUSION
The existing models ignore the effects of immediate surroundings of solid sample
to measure the thermal physical properties by photothermal spectroscopy. But our studies
show that there is significant effect of contact surroundings of solid sample to the thermal
expansion, surface temperature change, and the photothermal signal of sample.
Therefore, models that do not account for heat transfer to the surroundings are in error to
measure optical and thermophysical properties of solids by photothermal spectroscopic
methods. An improved approach to measure correct thermo- physical and optical
parameter of solid sample by thermal lens spectrometry has been presented.
Time resolved reflection photothermal lens experiment is carried out on Corning colored
glass filters and thermoelastic surface displacement is measured. FEA simulation of
thermal lens effect on glass coupling of heat both within the sample and out to the
surroundings for pulsed laser excitation is done. An experimental thermoelastic surface
displacement result is compared with that of modeling and are found to be very close.
The effect of the coupling fluid is incorporated into this model in order to minimize the
error in measuring physical constants.
REFERENCES
1. W. B. Jackson, N. M. Amer, A. C. Boccara, and D. Fournier, Appl. Opt. 20,
333(1981).
2. Olmstead, M. A, N.M. Amer, S.Kohn, D. Fournier, and A. C. Boccara., Appl. Phys.
A: Solid Surf. 32, 141 (1983).
80
3. Opsal, J, A. Rosencwaig, and D. L. Willenborg, Appl. Opt. 22, 3169 (1983).
4. Li, B. C.; J. Appl. Phys. 68, 482 (1990).
5. G. L. Bennis, R. Vyas, R. Gupta, S. Ang, and W. D. Brown, J. Appl. Phys. 84, 2602
(1998).
6. Bialkowski, S. E., Photothermal Spectroscopy Methohod of chemical analysis
(Wiley, New York, 1996).
7. J. C. Cheng, and S. Y. Zhang, J. Appl. Phys. 70, 7007 (1991).
8. P. K. Kuo, and M. Munidasa, Appl. Opt. 29, 5326 (1990).
9. J. W. Fang, and S. Y. Zhang, Appl. Phys. B: Lasers Opt. 67, 633 (1998).
10. B. Li, H. Blaschke, and D. Ristau, Appl. Opt. 45, 5827 (2006).
11. O. O. Dada,., M. R. Jorgensen, and S. E. Bialkowski, Appl. Spectrosc. 61, 1373
(2007).
12. N.G.C. Astrath, , L.C. Malacarne, P.R.B. Pedreira, A.C. Bento, and M.L.Baesso,
Appl. Phys. Lett. 91, 191908 (2007).
13. Francine B. G. Astrath, Nelson G. C. Astrath, Jun Shen, Jianqin Zhou, Luis C.
Malacarne, P. R. B. Pedreira, and Mauro L. Baesso, Optics Express 16, 12214 (2008).
14. N. G. C. Astrath, , F.B.G. Astrath, , J. Shen, C. E. Gu, L,. C. Malacarne, P.R.B.
Pedreira, A. C. Bento, and M. L. Baesso, Appl. Phys. B 94, 473 (2009).
15. F.Sato, L.C. Malacarne, P.R.B. Pedreira, M.P. Belancon, R.S. Mendas, M.L. Baesso,
N. G.C. Astrath, and J.Shen., J.Appl. Phys. 104, 053520 (2008).
16. P. R. Joshi, O. O. Dada, and S. E. Bialkowski, Appl. Spectrosc. 63, 815 (2009).
81
CHAPTER 5
PHOTOTHERMAL LENS SPECTROMETRY IN NANOLITER CYLINDRICAL
SAMPLE CELLS
ABSTRACT
A novel apparatus for performing photothermal lens (PTL) spectroscopy is
described which uses a low-volume cylindrical sample cell with pulsed excitation laser
and a continuous probe laser. The whole sample cell volume is irradiated with constant
irradiance beam produced by an excitation laser. The photothermal lens element is
formed by thermal diffusion from the irradiated sample volume through the sample cell
walls. The apparatus has been found to work with cells designed to contain sample
volumes to 361 nL. Larger and smaller volume cells are practical. However, the response
time increases with increasing sample cell radius. The theoretical thermal lens signal
increases with decreasing sample cell radius for a source with constant integrated
irradiance. Finite Element Analysis (FEA) modeling is used to examine the temperature
profile and the photothermal signal. The result of FEA is compared with the experimental
result.
INTRODUCTION
The technique of photothermal spectrometry has been used for low absorbance
measurements for several decades. It is an ultra-sensitive spectroscopy technique useful
for analytes that do not fluoresce. It has not, however, found widespread use in chemical
82
analysis owing, in part, to anomalous behavior arising from sample temperature
changes, changes in thermodynamic properties of the matrix, and nonlinear optical
effects.1
There are several physical effects that limit sensitivity and influence the accuracy
of measurements obtained using photothermal lens spectrometry.2-4 First, the
photothermal lens signal is related to the lens formed the sample as a consequence of
light absorption and subsequent energy transfer to the matrix. The optical element
produced when laser beams are focused into a homogeneous sample is not a simple lens.
The aberrant natures of the photothermal lens results in signal magnitudes that are
somewhat smaller than expected and also more difficult relate to sample absorbance. 5,6
Second, sample irradiation can produce large temperature changes. In theory, the on-axis
temperature change approaches infinity at long times for any excitation power when
using continuous laser excitation sources. In practice, large temperature changes distort
the thermal lens perturbation due to density differences and convection heat transfer or
even boiling may occur. 7-9 The later distorts the signal and ruins the analytical utility.
This paper described a novel apparatus that apparently circumvents many of
problems associated with photothermal spectrometry of homogeneous samples. The
apparatus uses a low-volume cylindrical sample cell, a chopped or pulsed excitation laser,
and a continuous probe laser. The full volume of the sample is irradiated with constant,
e.g., non-Gaussian, irradiance beam produced by the excitation laser. Constant irradiance
excitation source does not directly produce the photothermal lens element in the sample.
The lens element is formed by thermal diffusion from the irradiated sample volume,
83
through the sample cell walls. Under continuous irradiation, thermal diffusion results
in a parabolic temperature change profile.
The most important aspect of this experiment arrangement is that artifacts due to
excited state refractive index changes, volume changes, etc. do not affect the signal.
Although the refractive index change may depend on the partial refractive index and
partial molar volume of transient species, these changes do not result in a photothermal
lens element. By monitoring both the central portion and the full probe laser beam, the
apparatus can compensate for transmission changes due to bulk density, refractive index,
or absorbance changes.
This paper outlines the model used to relate the heat transfer and the photothermal
effect to experimental observation. Experiments to verify the operation of the apparatus
are performed with iron (II) dicyclopentadine (FeCp2) in ethanol. Photothermal lens
signals processed in the usual fashion are found to be relatively linear, reproducible, and
consistent with the model based on heat conduction through the sample cell walls. The
experimental photothermal lens enhancement is found to be that predicted from theory
within experimental error. FEA modeling has been also used for the better understanding
of temperature profile and for the comparison of the experimental results.
THEORY
In photothermal lens spectrometry, the measured signal is related to the focal
length of the lens formed when the sample absorbs optical radiation. Absorbed energy it
converted to heat. Sample heating changes the temperature and subsequently the
refractive index of the sample, producing the lens-like refractive index change. A probe
84
laser is used to detect the refractive index change. The theoretical signal is found by
first solving the thermal diffusion equation to determine the time-dependent
temperatures, second finding the thermal lens this temperature change produces, and
finally, calculating the signal that will be produced in the experimental apparatus.
Temperature change. The temperature change is found using the thermal
diffusion equation with the appropriate boundary conditions. For weakly absorbing
sample without significant attenuation along the z-axis, the radius, r (m), and time, t (s),
dependent thermal diffusion equation is
q (r , t )

T (r , t )  DT  r2T (r , t )  H
C P
t
1    
r 
 
r r  r 
2
r
In this equation, T (r, t) is the temperature change, DT (m2 s−1) is the thermal
diffusion coefficient, qH (r, t) (W) is the heat source term given as a rate of heat
production,  is density (kg m−3), and CP (J kg−1 K−1) is heat capacity.
This equation may be used to find an analytical solution when thermal diffusion
along the z-axis is negligible compared to that in the radial direction. This implies that the
sample is optically thin, i.e., low absorbance and that the cell windows are thermally
insulating relative to the cell walls. The later approximation is probably reasonable for
metal sample cells with glass windows, as is used here.
(1)
85
For illumination of the full sample cell volume, and when pulsed excitation and
excited state relaxation occurs much faster than thermal diffusion, the instantaneous heat
source term is independent of radius and the initial temperature change is
T0 
QYH
a 2 C P
(2)
Q (J) is the pulsed laser energy passing through the sample cell, α (m−1) is the exponential
absorption coefficient, and a (m) is the radius of the sample cell. The term YH is a unit
less heat yield term included to account for any radiative energy loss. It is the fraction of
energy absorbed and not lost by reemission of radiation. This term is unity for nonluminescent samples. The heat source term is thus
q H (r , t ) 
 QYH
 (t )
t a 2
(3)
(t) is a delta function. Since the thermal conductivity and heat capacity of the metal are
large relative to that of the sample, it is assumed that temperature change is zero at the
sample-cell wall interface. The time-dependent impulse-response for the temperature
change produced in a cylindrical cell with an internal radius, a (m), and zero cell wall
temperature is 10-11

T (r , t )  2T0  exp  n2 DT t / a 2 
n 1
J 0 r n / a 
 n J 1  n 
(4)
86
where δT0 is the instantaneous initial temperature change of the cylindrical region, J0
and J1 are the zero and first order Bessel's functions, and n is the n'th root of the Bessel's
function equation, J0(n)=0.
Photothermal lens signal. The photothermal lens strength is found from 1,12
d2
1
1  dn 
 
T (r , t ) ds

f (t )
2  dT  path dr 2
r 0
(5)
The term (dn/dT) is the thermal-optical coefficient. It relates the change in refractive
index to temperature change. Using the relationship |d2J0(ar)/dr2|r=0=-a2, and
substituting the integral by the path length, l (m), product, the time-dependent inverse
focal length is

n
1
 dn  l
2
2

 2 T0  exp   n DT t / a
f (t )  dT  a
J 1  n 
n 1


(6)
Using the initial temperature change results in the photothermal lens strength of
1
 dn  lQYH
=

f cyl (t )  dT  a 4  C P
 n exp  n2 t / 4t c 

J1 ( n )
n 1

The characteristic thermal time constant is defined as tc=a2/4DT.
The photothermal lens signal for the cylindrical sample cell is defined in the same
fashion as in the large-volume sample cell. 1 The time-dependent signal is found from
(7)
87
S (t ) 
 p (  )   p (t )
(8)
 p (t )
p(t) is the time-dependent probe laser power passing through a pinhole placed beyond
the sample cell. The reference at infinite time is used to indicate power of the unperturbed
probe laser, i.e., with no thermal lens. For a pinhole aperture placed far from the sample
cell, the signal is S(t)≈2z'/f(t), where z' (m) is the distance from the probe laser beam
focus position to the sample cell.
Model calculations exhibit an exponential response after a relatively short
induction period. The model behavior indicates that a single Bessel's root dominates the
time-dependent behavior. Examination of the model data shows that the first root,
1=2.405, dominates after an initial induction period. The inverse focal length is
approximated by
  t/ 4 t
1
 dn  lEY H 
e 1 c 

1
2




f cyl (t )  dT  2a 2  
1 J 1 (1 ) 
2
The characteristic time constant may be obtained from tc = 2.632t10%-90%, where
t10%90% is the time required for the signal to change from 10% to 90% of the maximum.
Thermal lens signal. The photothermal lens element is detected using a probe
laser. The thermal lens signal is defined as the relative change in optical power of a
probe laser beam passing first through the sample, and then through a small pinhole
(9)
88
aperture set to monitor the beam center and positioned far from the sample. The
photothermal lens strength is initially zero and increases with time. The maximum signal
is found from the relationship, S () = [Φp(0)-Φp()]/Φp(), where Φp(0) and Φp() are
the probe laser powers at zero and infinite times, respectively. Paraxial refraction theory
is used to determine the effect of the thermal lens on the probe laser propagation
characteristics. 13 When the pinhole aperture is placed far from the sample cell, the signal
is Scyl(t)=2z'/f(t), where Scyl(t) is used to signify the signal produced in the cylindrical cell
and z' (m) is the distance between the probe beam focus position and the sample cell.
The maximum thermal lens signal is thus
 dn  q H, path z 

S cyl (  ) = 
 dT   f
The practical probe laser focus distance, z', will be limited by the diverging probe
laser beam. Although this result was obtained using paraxial refraction, equivalent
results should be obtained using the diffraction optics approach since the limiting
photothermal lens element is parabolic in form. Note that the photothermal lens signal
does not depend on the spatial properties of the excitation beam, except through the
constant irradiance condition implied in the derivation. An exact signal equation for long
pathlength sample cells or high absorbance samples can be derived using the ray transfer
matrix for a quadratic profile medium.1
(10)
89
EXPERIMENTAL
A pulsed dye laser (Lambda Physik) operating at 490 nm is used as the excitation
source and a 632.6 nm HeNe laser (Uniphase, Model 1107P) is used to probe the
resulting photothermal signal. Collinear dual-beam geometry for the thermal lens
experiments is set up. The distance between the sample and the photodiode detector is
optimized to satisfy the far-field paraxial approximation. Two lenses (5 cm and 25 cm
focal length) are used to focus the excitation beam in the sample after the HeNe beam
focus.
The excitation source power is measured with a laser power meter. The
photothermal lens caused focusing and defocusing of the probe laser. This is measured as
a change in the power at the center of the beam. The center HeNe beam power is
measured using a pinhole and a United Detector Technology (UDT) Model PIN-10DP
photovoltaic photodiode detector. A 632.8 nm laser line bandpass filter is used to prevent
the transmitted dye laser beam from being detected by the photodiode detector. A small
fraction of the probe beam is split off prior to the pinhole aperture and a second bandpass
filter/photodiode is used to monitor the probe laser power past the sample. Changes in the
probe laser power are compensated for using an operational amplifier divider, the circuit
of which divides the thermal lens signal by the signal proportional to the HeNe laser
power. This probe laser power compensated thermal lens signal is amplified and
electronically filtered with a Tektronix model AM-502 differential amplifier. The analog
signal is subsequently digitized with a 16-bit analog-to-digital converter board and
processed by multichannel analysis software.
90
Figure 5-1. A schematic drawing of a photothermal lens experimental setup.
The latter averages several signal transients. Multichannel averaging was performed to
improve the raw photothermal lens signal estimation precision. The photothermal lens
signal was calculated from this raw data using a simple spreadsheet program. The
diagram in Figure 5-1 illustrates the apparatus setup for the thermal lens experiment.
Sample cells and sample. A cylindrical sample cell made of fused silica glass
having an internal diameter 240 µm and external diameter 1000 µm and fitted in 1 cm3
steel block was used for the study. Cylindrical sample cell results were compared to those
obtained using a conventional 1 cm pathlength spectrophotometry cell (cuvette). Ethanol
FeCp2 solutions were used as liquid samples for the study. Linear dilution was used to
obtain lower absorbance from stock solutions of high enough absorbance to measure
using a spectrophotometer with a 1 cm cuvette. The sample was positioned at the focus of
the excitation beam for maximum temperature gradient and the cylindrical sample cell
91
versus conventional cell experiment was carried out separately at room temperature
under the same conditions. Sample absorbances were recorded with a Cary 3E UVVisible Spectrophotometer.
Finite element analysis. Finite element analysis software provides numerical
solutions to the heat transfer equations with the realistic boundary conditions imposed by
the complicated experimental geometries. To better understand the transient temperature
profile in the samples, finite element analysis is used to model stationary temperature
changes. The result of the finite element analysis is then compared to conventional
analytical solutions to gauge the error. The experimental setup and the apparatus
constraints are guided by the error analysis. Analysis based on Comsol Femlab v3.3 is
carried out on a personal computer (Compaq Presario SR1330x, AMD Athlon XP 3200
processor).
The analysis consists of the drawing the sample geometry; specify material
boundary conditions, heat sources, and sinks. And then the problems are solved with
rough finite element definition and further refinement of elements and domain are made.
Finally, dT can be obtained either at single time, a time series or at steady state. The
relative photothermal lens signal strength is found from Eq. 5.The dn/dT for ethanol is
taken as -4x10-4 K-1. The path integral of the second radial derivative was found by using
the Comsol Integration coupling variable to integrate the 2nd-derivative function of the
temperature change. The relation S(t) = 2z’/f(t) is used to generate the theoretical signal.
A cylindrical sample cell made of silica glass having an internal diameter 240 and
external diameter 1000 µm of 8 mm height, fitted in 1 cm3 steel blocks filled with ethanol
was modeled using Comsol Multiphysics to represent the actual sample cell used in the
92
experiment. For convenience, the cell was modeled with 2D axial symmetry the origin
at the center that the z-axis runs along the path length of the excitation and probe beams.
The temperature profile of the ethanol (containing FeCp2) in cylindrical cell was obtained
by having the FEA software solve the heat equation with the boundaries of glass and steel
set at zero degrees and heat input along the z-axis defined by Eq. 4. This way, the
temperature solution represents dT. The model was solved using steady-state conditions.
The values of α, Q, and w used were 2.1 (m-1), 0.167 (mW), and 45 (μm), respectively.
RESULTS AND DISCUSSION
Figure 5-2 shows 100 averaged signals obtained using 2.1 AU m-1 FeCp2 ethanol
solutions in cylindrical and standard cells. The time-dependent response and signal
magnitude differences are expected due to the thermal diffusion boundary conditions.
Maximum signals for cylindrical and standard cell experiments are 0.31 and 2.65. The
different signal magnitudes are expected from the theoretical signal ratio
Scylindrical / Sstandard = w2 / 2a2
w (m) is the laser beam waist radius and a(m) is cylinder radius. The experimental
signal response time for cylindrical and standard sample cell is 0.011 s and 0.0057 s
respectively. This slower response time is expected for cylindrical sample cell.
The time constants of cylindrical and standard sample cell PTS are 0.079 and 0.128
second respectively (t10%-t90%-method). Using the thermal diffusion coefficient of 8.9×108
m2s-1 for ethanol, cylindrical cell radius of 167 µm is obtained from the characteristic
time constant for cylindrical cell photothermal signal.
93
3.0
Standard-cell
Cylindrical-Cell
2.5
Signal
2.0
1.5
1.0
0.5
0.0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Time (s)
Figure 5-2. Photothermal lens signal of standard and cylindrical sample cell of FeCp2 in
ethanol.
This is in reasonable agreement with the 120 µm measured radius of the sample cell.
FEA modeled signals for 60-200 µm radii are presented in figure 5-3. There is
variation of signal magnitude as well as the time constants for the signals of theses sizes.
The Comsol multiphysics modeling signal versus cylindrical sample cell radius plot in
the Figure 5-4 supports the theory that the signal magnitude increase with decreasing
radii of cells. Signal produced from the FEA modeling of 120-micron size cell and that of
the experimental signal of the same size in the figure 5-5 shows that they have reasonably
close in magnitude (0.62 & 0.31). From modeling it is sown that (Figure 5-6, 5-7) the
signal response time and time constant are directly related with the radii of cells.
94
3.0
60um
80um
2.5
100um
120um
Signal
2.0
140um
160um
180um
1.5
200um
1.0
0.5
0.0
0
0.01
0.02
0.03
0.04
0.05
Time (s)
Figure 5-3. FEA modeling PTL signals of cylindrical sample cells of 60 to 200 µm
sizes.
6
5
Signal
4
3
2
1
0
0
50
100
150
Cell radius (um)
Figure 5-4. Variation of signal magnitude with sample cell radius.
200
250
95
0.7
FEA modelling Signal
0.6
Experimental Signal
Signal
0.5
0.4
0.3
0.2
0.1
0
0.00
0.05
0.10
0.15
0.20
Time (s)
Figure 5-5. FEA modeling and experimental PTL signal in 120-µm radius sample cell
of FeCp2 in ethanol solution.
0.06
0.05
Time (s)
0.04
0.03
0.02
0.01
0
0
50
100
150
200
Cell radius (um)
Figure 5-6. Variation of signal rises time with sample cell radius.
250
96
0.6
Time constant (s)
0.5
0.4
0.3
0.2
0.1
0
0
50
100
150
200
250
Cell radius (um)
Figure 5-7. Variation of time constant with cell radius.
0.6
0.5
Time (s)
0.4
0.3
0.2
0.1
0
0
5000
10000
15000
20000
25000
30000
35000
40000
Square of cell radius (um2)
Figure 5-8. Variation of time constant with square of the cell radius.
45000
97
Figure 5-8 is variation of time constant with square of the cell radius defined by the
equation tc=a2/4DT.
A signal magnitude of 0.31 is obtained for a sample with an absorbance of 0.021
this corresponds to a signal enhancement of 14.76. 1 The theoretical enhancement for the
4.4e-4J energy used is 123 this discrepancy is not surprising. The maximum theoretical
enhancement is rarely realized due to the dependence on optical design parameters such
as the probe laser focus position and distance to the pinhole detector.
CONCLUSION
A novel apparatus for performing photothermal lens (PTL) spectroscopy has been
described which uses a low-volume (361nL) cylindrical sample cell with pulsed
excitation laser. And FEA modeling is used to examine the temperature profile and the
photothermal signal.
The advantages of apparatus over conventional cell are that it requires a little
sample, and signals produced by thermal diffusion through the sample wall are less
susceptible to the bulk heat transfer effects due to convection. The finite on axis
temperature change, when low energy pulsed laser is used, prevents the solvent from
boiling and prevent signal instabilities due to turbulent convection heat transfer. The
cylindrical sample cell circumvents the large temperature changes by heat transport to the
surrounding. The parabolic-form photothermal lens produced by thermal diffusion is
aberration free. Subsequently, the beam propagation theory that only approximately
describes the effect of the photothermal lens produced from laser excitation exactly
98
describes the effect of the cylindrical cell lens. Although the refractive index change
may depend on the partial refractive index and partial molar volume of transient species,
these changes do not result in a photothermal lens element. By monitoring both the
central portion and the full probe laser beam, the apparatus can compensate for
transmission changes due to bulk density, refractive index, or absorbance changes.
The experimental photothermal lens enhancement has been found to be that
predicted from theory within experimental error. Further study to optimize the apparatus
and the experiment is required for the best performance.
REFERENCES
1. S. E. Bialkowski, Photothermal Spectroscopy Methods for Chemical Analysis
(Wiley, New York, 1996).
2. N. J. Dovichi, and J. M. Harris, Anal. Chem. 51, 728 (1979).
3. K. Mori, T. Imasaka, and N. Ishibashi, Anal. Chem. 55, 1075 (1983).
4. C. A. Carter, and J. M. Harris, Anal. Chem. 55, 1256 (1983).
5. C. A. Carter, and J. M. Harris, Appl. Opt. 23, 476 (1984).
6. J. Shen, R. D. Lowe, and R. D. Snook, Chem. Phys.165, 385 (1992).
7. N. J. Dovichi, and J. M. Harris, Anal. Chem. 53, 106 (1981).
8. C. E. Buffett, and M. D. Morris, Appl. Spectrosc. 37, 455 (1983).
9. E. F. Simo Alphonso, M. A. Rius Revert, M. C. Garcia Alvarez-Coque, and G.
Ramis Ramos, Appl. Spectrosc. 44, 1501 (1990).
10. A. Chartier, and S. E. Bialkowski, Opt. Eng. 36, 303-311 (1997).
99
11. H. S. Carlaw, and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed.
(Clarendon Press, Oxford, 1959).
12. W. B. Jackson, N. M. Amer, A. C. Boccarra, and D. Fournier, Appl. Opt. 20,
1333–1344 (1981)
13. A. Yariv, Optical Electronics, 3rd ed. (CBS College Publishing, New York,
1985).
100
CHAPTER 6
ANALYTICAL SOLUTION FOR MODE-MISMATCHED THERMAL LENS
SPECTROSCOPY WITH SAMPLE-FLUID HEAT COUPLING b
ABSTRACT
This paper presents an improved theoretical description of the mode-mismatched
thermal lens effect using models that account for heat transport both within the sample
and out to the surrounding coupling medium. Analytical and numerical finite element
analysis (FEA) solutions are compared and subsequently used to model the thermal lens
effect that would be observed using continuous laser excitation. FEA model results were
found to be in excellent agreement with the analytical solutions. The model results show
that heat transfer to the air-coupling medium introduces only a minor effect when
compared with the solution obtained without considering axial air-sample heat flux for
practical examples. On the other hand, the thermal lens created in the air coupling fluid
has a relatively more significant effect on the time-dependent photothermal lens signals.
INTRODUCTION
Thermal lens (TL) spectroscopy is a remote, nondestructive, fast, and highly
sensitive photothermal technique for the measurement of optical absorption and thermooptical properties of materials.1-13 This technique has been applied to obtain optical and
b
COAUTHORED BY PR JOSHI, LC MALACARNE, NGC ASTRATH, PRB PEDREIRA, RS MENDES, ML
BAESSO AND SE. BIALKOWSKI. REPRODUCED WITH PERMISSION FROM THE JOURNAL OF APPLIED
PHYSICS, 107, 053104 (2010) COPYRIGHT 2010 AMERICAN INSTITUTE OF PHYSICS (SEE APPENDIX C)
101
thermal-optical properties of a wide range of material, including glasses, oils,
polymers, and liquid crystals.7–13 TL technique has also been demonstrated as a powerful
method for chemical analysis.4,8
Theoretical analytical solutions restricted the use of TL spectroscopy to low optical
absorbing samples. Recently, a simple approximation was proposed by taking the high
optical absorbing material case into account14 and also the complementary thermal mirror
method15–18 was developed to use concurrently with TL leading to additional information
about physical properties of high and low absorbing samples. Despite numerous
applications and theoretical descriptions, TL model has no analytical solution that
considers the axial heat coupling between sample and external medium into account.
Recently,19–21 by using finite element analysis (FEA), it has been demonstrated that the
effect of surface heat transfer from sample to the surroundings could introduce some
modification in the physical parameters measured by pulsed laser excited photothermal
lens spectroscopy.
In this work, we present a semi-analytical theoretical description of the modemismatched TL effect by taking the coupling of heat both within the sample and out to
the surroundings into account. Our analytical results are compared with FEA solutions.
These two methods for calculating the time- and space-dependent heat transfer are in
excellent agreement lending some degree of confidence in the predictions of these
solutions. We show that the interface effect occurs in finite characteristic lengths. For
samples with thickness larger than this characteristic length, a simplified solution can be
used to obtain the TL phase shift within the sample and surrounding fluid.
102
THEORY
A typical configuration used in mode-mismatched TL spectrometry is illustrated in
Fig. 6-a. A continuous Gaussian excitation laser beam irradiates a weakly absorbing
sample of thickness l, causing a TL. A second, often-weaker Gaussian beam propagates
through the sample collinear to the excitation laser and is affected by the TL. The
characteristic electric field radii of the excitation and probe beams in the sample are 0e
and 1P , respectively. The probe beam propagates in the +z-direction, and the sample is
centered at z=0. The distance between the sample and the detector plane is Z2 and the
distance between the sample and the minimum probe beam waist of a radius w0P is Z1. In
this configuration, it is assumed that (i) the sample dimensions are large compared to the
excitation beam radius to avoid edge effects; (ii) the absorbed excitation laser energy by
the sample is low so that the excitation laser can be considered to be uniform along the zdirection.
Temperature gradient. To model the sample-fluid heating coupling, let us
consider two semi-infinity spaces with boundaries at z=0 with the sample in the 0<z<∞
region and the fluid (air in this work) in the ∞<z<0 region. The temperature rise
distributions inside the sample, Ts(r , z , t), and in the fluid, Tf (r , z , t), are given by the
solution of the heat conduction differential equations
Ts  r , z , t 
t
T f  r , z, t 
t
 D 2Ts  r , z , t  = Q  r , z 
(1)
 D f  2T f  r , z , t  = 0
(2)
103
Figure 6-a. Scheme of the geometric positions of the beams in a mode-mismatched dual-
beam TL experiment.
Figure 6-b. sample geometry used for the finite element analysis modeling.
104
The boundary and initial conditions are given by
k  zTs r , z , t  /  z
z0
 k f  zT f
r , z , t  /  z
z0
T s  , z , t   T s r ,  , t   0
Ts r , z , 0   T f
r , z , 0   0
Ts r , 0 , t   T f r , 0 , t 
T f  , z , t   T f r ,   , t  
Di  ki / i ci
(3)
0
is the thermal diffusivity of the material i (sample and fluid). ci , i and ki are
the specific heat, mass density, and thermal conductivity of the material i , respectively.
The Gaussian excitation laser profile is Q  r , z   Q0 Q  r  Q  z  , where
Q  r   exp  2r 2  2  , Q0 = 2 Pe Ae  c  2 , Pe
is the excitation beam power, Ae is the optical
absorption coefficient at the excitation beam wavelength e ,  = 1  e / em ,where em is
the average wavelength of the fluorescence emission, and  is the fluorescence quantum
efficiency, which competes for a share of absorbed excitation energy. Q(z)=exp(Ae z) is
the sample absorptance. For the purposes here, the optical absorption coefficient is small
and Q(z)=1 can be assumed.
Using Laplace and Hankel integral transform methods with the boundary conditions
in Eq. (3), the solution of the heat conduction differential equations in the Laplace–
Hankel space can be written as


s  Df  2 ez  sD  D
Q  
Ts , z, s  

 k D s  D  2  s  D 2 s  s  D 2 
f
f
2 

s  s  D  1
 k D s  D 2 
f


Q  k f D k Df
2
(4)
105
and
Tf  , z, s   
Q    s  Df  2
 k D s  D 2
f
f
s  s  D  1 
 k D s  D 2
f

2
Here Q    Q0 exp   2 2 8   2 4
z
e




 sD   D
f
2
f
s  Df  2
(5)
is the Hankel transform of the source term.
If k f D s  D f  2  k D f s  D 2 , the term in the denominator of Eqs. (4) and (5) can
1
be expanded in series as 1  x   1  x  x 2  O  x3  . This condition is fulfilled in the case of
glass-air system. Thus, Eq. (4) can be written as
 e  z  s  D 2  D
s  Df  2  k f D s  Df  2
Q  
1 
 

Ts  , z , s   Q  
2
 s  D 2
s  s  D 2 
k D f s  s  D  
k D f s  D 2


kf
D
(6)
and Eq. (5) becomes
 e z  s  D f  2  D f
s  Df  2  k f D s  Df  2

T f   , z , s   Q   
1
 
2
 s  D 2
s  s  D 2  
k
D
s
D


f
f


The time-dependent temperatures are given by the inverse Laplace and Hankel
transforms of Eqs. (6) and (7). Taking the expansion in Eqs (6) and (7) up to the first, the
temperature gradient within the sample, T1(s) (r , z , t), is given by the integral equation
(7)
106
T1 s  r, z, t   Q0 
t


e
2 2 8

4
0 0

kf D
k Df
e
2


 J0  r  e D
 z2
4 D t  
 D erf
 f
  t   

e
 D t   2

2

 Df  D  e D
2
Df  D erf

D

 Df  D 




(8)
d d
while that for the coupling fluid, T1 f   r , z, t  , is
T1 f   r, z, t   
t


0 0
2 2 8

e
4
2
Df  t  
e
2
 z2
4Df  t  
 Df erf

  t   

e



2

 Df  eD Df  D erf  Df  D
D


2
k f D  2eD   D  DF  erf  D Df 


 J r  d d .
 0
D3/ 2
k Df 
D 


(9)
J n  x  is the Bessel function of the first kind and erf  x  is the error function.
Considering the zero order expansion, which in fact correspond to the case where
there is no heat flux from sample to air, or  z T  r , z, t  / z
T0  s  ( r , t ) =
z 0
 0 , Eq. (8) becomes
Q0


 2 r 2 / 02e
exp
0  1  2 /tc     1  2 /tc


t

 d

(10)
Eq. (10) is the temperature gradient commonly used to describe the Thermal Lens and
Thermal Mirror effects for low absorbing semitransparent solids.9,15-17 In addition, the
zero order approximation for the air temperature solution, which correspond to the
107
solution for a semi-space system with the temperature in z  0 surface fixed by the
sample's temperature, is given by
T0 f   r, z, t   
t


0 0
2 2 8
Df  t   2
4e
 z2
4Df  t  
 Df erf

  t   

 e
e
2



  J r d d.
 Df  eD Df  D erf  Df  D 
2
D


0
(11)
Finite element analysis. In order to ascertain the accuracy of our approximation
the space and time dependences of the temperature equations for air and sample in Eqs.
(8) to (11) are compared with the FEA modeling solutions for a system with glass sample
of thickness l surrounded by air. FEA software provides numerical solutions to the heat
transfer equations with the realistic boundary conditions imposed by the experimental
geometry. To better understand the transient temperature profile in the samples, FEA is
used to model temperature changes. Results from the FEA calculations are then
compared with the semi-infinity analytical solutions to gauge the error. COMSOL
MULTIPHYSICS 3.5 analysis is carried out on a Dell Studio XPS 435, i7 940 processor,
using MS WINDOWS VISTA. The numerical integrations in Eqs (8)-(11) are performed
using the standard functions in MATHEMATICA (version 7.0) software.
The COMSOL MULTIPHYSICS software in conduction and convection mode
solves the heat diffusion equation given as
C p

T (r , z, t )  k 2T (r , z, t )  q H (r , z, t )  C p u  T (r , z, t )
t
(12)
108
in which u is the flow velocity. All other symbols are the same as those introduced
above. Note that Eqs. (1), (2), and (12) differ only by the second term on the right side of
the equation. This term can account for convection or mass flow heat transfer, which is
not important for glasses.
FEA modeling consists of drawing the sample geometry and specifying material
boundary conditions, heat sources, and sinks. The problems are then solved with rough
finite element definition and further refinement of elements and domain are made. The
element mesh is refined until model results become independent of mesh size. Finally,
T(r, z , t) can be obtained either at a single time, over a time series, or at steady state. The
model solid absorbing media was cylindrical plate of 10 mm in diameter and 1 mm thick.
Optical excitation was along the z-axis. The values of the thermal, optical, and
mechanical parameters used for the analytical and FEA modeling simulations are shown
in Table 6-1.
Figure 6-2 shows the radial temperature profile within the glass sample at its center
at z=0.5 mm. The results by the FEA model are compared with the temperature solution
considering heat flux from glass to air, Eq. (8), at different exposure times. There is
apparently little difference between the temperatures produced from the FEA model and
the semi-infinite approximation with glass-air heat coupling.
Figure 6-3 shows the temperature profiles along the z-direction (air-glass-air) using
the FEA modeling, the solution considering air surroundings, Eq. (8), and the solution
with no transfer of heat from glass to air, Eq. (10), at different exposure times at r=0. It
clearly shows an excellent agreement between the analytical solutions considering aircoupling Eqs. (8) and (9) and the FEA model. On the other hand, as expected, there is a
109
little difference for the predictions considering no heat transfer from glass to air, Eq.
(10). This difference is more evident close to the boundary sample-air.
Table 6-1. Parameters used for the simulations. The thermal, optical and mechanical
properties listed below are associated to characteristics values found in glasses
12,18
for the surrounding medium we use the air properties. 22
Pe
(mW)
161
f
(kg/m3)
1.18
Ae
(m-1)
93
cf
(J/kgK)
1005
D
(10-7 m2/s)
5

(m)
50
k
(W/mK)
1.4
tc   2 4 D
(ms)
1.25

(kg/m3)
933

c
(J/kgK)
3000
T
Df
(10-5 m2/s)
2.2

kf
(W/mK)
0.026
Q0
ds dT
(10-6 K-1)
10
 dn
60
V
m
0.6
(10-6 K-1)
7.5
0.25
dT  f
(WK s-1)
818. 80356
(10-6 K-1)
-1
5
and
110
Figure 6-2. Temperature profile in the glass sample at z  0.5 mm using the FEA
modeling, the solution considering heat flux to air, Eq. (8), and the solution with no
transfer of heat from glass to air, Eq. (10), at different exposure times.
Figure 6-3. Temperature profile along the z direction (air-glass-air) using the FEA
modeling, the solution considering air surroundings, Eqs.(8) and (9), and the solution
with no transfer of heat from glass to air, Eq. (10), at different exposure times. r  0 was
used in the simulations.
111
Figure 6-4. Temperature profile along the z direction (air-glass-air) using the FEA
modeling, the solution considering air surroundings, Eqs. (8) and (9), and the solution
with no transfer of heat from glass to air, Eq. (10), for different radial positions.
Figure 6-5. Radial temperature profile in the air surroundings at t  0.12 s using the FEA
modeling, the solution for the air fluid, Eq. (9), at the sample surface in the air side, z  0 ,
and up to 200 m distant from there.
112
(a)
(b)
Figure 6-6. Density plot using (a) the FEA and (b) the analytical approximation models.
In the case of glass-air interface, the characteristic length is around z=0.2 mm into
the sample. Deeper than that the temperature is the same as that one with no axial heat
fluxes as long as the excitation beam waist is small relative to sample thickness. This
shows that the interfaces do not affect each other if the sample is thick relative to the
beam waist. Subsequently, the single-surface approximation used for the analytical model
is valid for these conditions.
The same behavior is also observed for different radial positions, as shown in Fig.
6-4 at t=0.12 s. It is interesting to note the excellent agreement between FEA modeling
and the analytical solution, Eqs. (8) and (9).
Figure 6-5 displays the radial temperature profile in the air using the FEA
modeling and the solution for air, Eq. (9), at the sample surface in the air side from z=0 to
200 m. The exposure time was t=0.12 s. Again, agreement between both the FEA and
113
analytical modeling predictions is good. The results obtained using the FEA model
show that the first order analytical approximation presented in this work can be used to
describe the temperature profile in the sample and in the air surrounding it, at least in
cases where binomial expansion is valid. In addition, the order zero approximation also
represents quite well the temperature in the sample for thick samples. The density plot
illustrated in Fig.6- 6 shows the complete solution for both the FEA and analytical
approximation models within both air and sample.
PROBE BEAM PHASE SHIFT AND THERMAL LENS INTENSITY
The temporal and radial distribution of the temperature rise inside the sample and
in the air induce a refractive index gradient, acting as an optical element, causing a phase
shift  to the probe beam. The solutions presented in Eqs. (8) and (9) can be analytically
used to describe theoretically the TL effect, by calculating the phase shift of the probe
beam after passing thought the TL region and, therefore, to express the TL intensity
signal.
For low optical absorbing samples, we can use our semi-infinite solution for the
temperature, given by Eq. (8), to obtain the phase shift for finite sample of thickness l by
the following equations
s  r , t  =
2 ds l 2
2 T  r, z, t   T  0, z, t   dz
P dT 0 
(13)
114
where the integration is from 0  z  l / 2 , however multiplied by a factor 2 .ds/dT is
the temperature coefficient of the optical path length at the probe beam wave length  p
For the air domain.
 f  r, t  =
0
2  dn 

 2  T f  r , z, t   T f  0, z, t   dz
P  dT  f
(14)
Here the factor of 2 accounts for the layers of air on both sides of the sample and
 dn
dT  f
is the temperature coefficient of the refractive index of the air. The total phases
shift,   r , t    ( s )  r , t    ( f )  r , t  describes the distortion of the probe beam caused by the
temperature change in the medium.
Substituting Eq. (8) into (13), and performing the z integration, one gets the first
order phase shift in the sample as
1 s  g, t   s 

0
 D 2
e

2 2

e  8 2 D k f D t   Dt   2 
erf l 4 D t  
e


4
 D k Df 0 
Df  D erf


Df erf

e  82 l  D 2  
e
 Df  D  d 
 J0  mg 1 d

4
2


2 2



 Df 
(15)
Alternatively, Eq. (9) into (14), and performing the z and the time integrations, one gets
the first order phase shift in the air as
115
1 f   g , t    f


0
2 2

e   8 2  D f
 D
4




2
 erf t D f
e  Dt t erfi t D  D f



 2 Df
 2D D  Df

 Dt 2
t  erf
k f D  2 D  e
 Df
D 3 / 2 3
k Df 



  


t D  
    J  mg  1 d

  0
 

(16)

erfi  x  is the imaginary error function, g =  r/1P  , m = 12P / 2 ,  f   4  p   dn dT  f Q0 and
2
 s   4  p   ds dT  Q0 . Using Esq. (10) and (11) one can write the zero order
approximation for the phase shift in the air and sample as
 0 s   g , t    s

 2mg 2 
l 2 
2
2
 Ei  2 m g   Ei   2
  Log    Log   8 Dt  
16 D 
   8 Dt 

(17)
in which Ei( x) is the exponential integral function, and
 0 f   g , t    f


0
2 2

e   8 2  D f
 D
4




2
 erf t D f
e  Dt t erfi t D  D f


  2 Df
 2 D D  Df

   J
 
 
0


mg  1 d

(18)
Eq. (17) is the phase shift commonly used to describe the Thermal Lens effect 7 for low
optical absorbing solids.
The complex electric field of a TEM00 Gaussian probe beam emerging from the
sample can be expressed as 3
   r2
 r2 
   2 ,
U P  r , Z1  = B exp  i 
  P R1P
 1P 
(19)
116
with B = 1P1 2 PP / exp  i 2 Z1 /P  . PP and R1P are the probe beam power and the radius of
curvature of the probe beam at Z1 . The propagation of the emerged probe beam from the
sample to the detector plane can be treated as a diffraction phenomenon. Using Fresnel
diffraction theory, its complex amplitude at the detector plane can be obtained as
described in Ref. 3. In this work only the center point of the probe beam at the detector
plane is considered. Then, the complex amplitude of the probe beam at the centre, using
cylindrical coordinates, is given by 3

U ( Z1  Z 2 , t ) = C  exp   1  iV  g  i ( s ) ( g , t )  i ( f ) ( g , t )  dg
(20)
0
when Z 2  Z c . Here V = Z1 /Z C , Z C is the confocal distance of the probe beam,
and C = B exp  i 2 Z 2 /P  i12P /P Z 2 . Substituting Eqs. (15)- (18) Into Eq. (20) and carrying
out numerical integration over g , the intensity I  t  at the detector plane can be calculated
as I  t   U  Z1  Z 2 , t  .
2
The effect of the TL on the probe beam is only to induce a phase shift. The
geometrical configuration of the probe and excitation beams defines the sensitivity of the
TL method by means of m . This parameter can be modified either by changing 1 p or  .
When r  1 p , within which more than 86% of the probe beam power is included, the
time dependence of the phase shift, assuming g  1 , is  1,t  . It can be calculated using
Eqs. (15) or (17) for the sample and Eqs. (16) or (18) for the air. The phase shifts
117
calculated using the parameters of Table 6-1 with g=1 is shown in Fig. 6- 7. Here the
time scale is in units of tc. One can see from Fig. 6-7 that the zero-order approximation
for the phase shift created in the sample agrees with the first-order one. The relative
difference is less than 0.3%. The air phase shift it is approximately 1.7% of that of the
glass sample.
The relative difference between phase shifts calculated with zero- and first-order
approximations is more important when the sample thickness is reduced. Figure 6-8
shows this difference at t=0.12 s as a function of the sample thickness. With a thin
sample, boundary effects become important and the first order approximation should be
used.
A ir P h a s e S h ift -  1(f) (1,t)
0 .0 0 4
A ir P h a s e S h ift -  0(f) (1,t)
0 .0 0 3
1 .8
0 .0 0 1
A ir
0 .0 0 0
0 .0 0
G la s s
-0 .0 5
1 .5
Relative error (%)
Phase shift, (1,t)
0 .0 0 2
0 .9
A ir
S a m p le
0 .6
0 .3
0 .0
-0 .1 0
R e la tiv e e rro r b e tw e en
 1  an d  0
1 .2
0
50
1 00
150
t/t c
-0 .1 5
-0 .2 0
S am p le P h a s e S h ift -  1(s ) (1 ,t)
-0 .2 5
S am p le P h a s e S h ift -  0(s ) (1 ,t)
0
20
40
60
80
100
t/t c
Figure 6-7. Probe beam phase shift in air and in the sample calculated using order 0 and
1 approximations as a function of time. The inset shows the relative error between both
approximations.
118
1 .2
Relative error (%)
1 .0
R e la t iv e e r r o r b e t w e e n
 1 (s )a n d  0 (s )
0 .8
0 .6
0 .4
t = 0 .1 2 s
0 .2
0 .0
0 .0
0 .5
1 .0
1 .5
2 .0
l(m m )
Figure 6-8. Relative difference between the zero and first order approximations for the
sample phase shift at t  0.12 s as a function of the sample thickness.
Normalized TL signal (au)
1 .1 2
1 .1 0
1 .0 8
1 .0 6
1 .0 4
A ir a n d S a m p le C o n t r ib u t io n s - O r d e r 1
S a m p le C o n t r ib u t io n ( N o h e a t f lu x )
A ir a n d S a m p le C o n t r ib u t io n s - O r d e r 0
1 .0 2
1 .0 0
0 .0 0
0 .0 2
0 .0 4
0 .0 6
0 .0 8
0 .1 0
0 .1 2
T im e ( s )
Figure 6-9. Normalized TL signal calculated using the approximations and the
parameters listed in Table 6-1. The sample thickness used was l  1mm .
119
Finally, Fig. 6-9 shows the TL signal calculated using the analytical
expressions and the phase shift. The agreement between the zero and first order
approximations is again good when both the TL contributions from the air and the sample
elements are taken into account. For the case where no air effect is included, the intensity
transient deviates a little from the expected using the air coupling solution. This
difference could lead to an overestimation of the thermal diffusivity and the parameter θ
of approximately 2%.
CONCLUSION
We presented analytical solutions for the temperature gradient induced by the TL
effect considering both the heat transfer from sample to air and the TL generated in the
air surroundings. Our analytical solution was found to agree with that of our FEA
software. The results showed that heat transfer between the sample surface and the air
coupling fluid does not introduce an important effect in the optical phase shift when
compared with the solution obtained without considering the air-sample heat flux. On the
other hand, when the TL created in the air coupling fluid is taken into account, a
significant effect is introduced on the predicted time-dependent TL signals, which
corresponds to approximately 1.7% of the sample’s TL effect. The effect is due to finite
heat transfer at the interface typically neglected in semi-infinite cylinder approximations.
These solutions open up the possibility of applying the TL method for accurate prediction
of the heat transfer to the coupling fluid and subsequently to study the gas surrounding
the samples by using a known material solid sample.
120
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1.
J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, J.
Appl. Phys. 36, 3 (1965).
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R. D. Snook, and R. D. Lowe, Analyst 120, 2051 (1995).
3.
J. Shen, R. D. Lowe, and R. D. Snook, Chem Phys. 165, 385 (1992).
4.
S. E. Bialkowski, Photothermal Spectroscopy Methods for Chemical Analysis
(Wiley, New York, 1996).
5.
A. Mandelis, ed., Progress in Photoacoustic and Photothermal Science and
Technology (Elsevier, New York, 1991).
6.
D. P. Almond, and P. M. Patel, Photothermal Science and Techniques (Chapman
& Hall, London, 1996).
7.
M. L. Baesso, J. Shen, and R. D. Snook, J. Appl. Phys. 75, 3732 (1994).
8.
M. Franko, and C. D. Tran, Rev. Sci. Instrum. 67, 1 (1996).
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J. Shen, M. L. Baesso, and R. D. Snook, J. Appl. Phys. 75, 3738 (1994).
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M. L. Baesso, A. C. Bento, A. A. Andrade, J. A. Sampaio, E. Pecoraro, L. A. O.
Nunes, T. Catunda, and S. Gama, Phys. Rev. B 57, 10545 (1998).
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S. M. Lima, T. Catunda, R. Lebullenger, A. C. Hernandes, M. L. Baesso, A. C.
Bento, and L. C. M. Miranda, Phys. Rev. B 60, 15173 (1999).
12.
E. Pelicon, J. H. Rohling, A. N. Medina, A. C. Bento, M. L. Baesso, D. F. de
Souza, S. L. Oliveira, J. A. Sampaio, S. M. Lima, L. A. O. Nunes, and T. Catunda,
J. Non-Cryst. Solids, 304, 244 (2002).
121
13.
N. G. C. Astrath, J. H. Rohling, A. N. Medina, A. C. Bento, M. L. Baesso, C.
Jacinto, T. Catunda, S. M. Lima, F. G. Gandra, M. J. V. Bell, and V. Anjos, Phys.
Rev. B 71, 14202 (2005).
14.
M. P. Belancon, L. C. Malacarne, P. R. B. Pedreira, A. N. Medina1, M. L.
Baesso, A. M. Farias, M. J. Barbosa, N. G. C. Astrath, and J. Shen, Submitted to J.
Physics: Conference Series (2009).
15.
N. G. C. Astrath, L. C. Malacarne, P. R. B. Pedreira, A. C. Bento, M. L. Baesso,
and J. Shen, Appl. Phys. Lett. 91, 191908 (2007).
16.
L. C. Malacarne, F. Sato, P. R. B. Pedreira, A. C. Bento, R. S. Mendes, M. L.
Baesso, N. G. C. Astrath, and J. Shen, Appl. Phys. Lett. 92, 131903 (2008).
17.
F. Sato, L. C. Malacarne, P. R. B. Pedreira, M. P. Belancon, R. S. Mendes, M. L.
Baesso, N. G. C. Astrath, and J. Shen, J. Appl. Phys. 104, 053520 (2008).
18.
N. G. C. Astrath, F. B. G. Astrath, J. Shen, J. Zhou, P. R. B. Pedreira, L. C.
Malacarne, A. C. Bento, and M. L. Baesso, Opt. Lett. 33, 1464 (2008).
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O. O. Dada, M. R. Jorgensen, and S. E. Bialkowski, Appl. Spectrosc. 61, 1373
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P. R. Joshi, O. O. Dada, and S. E. Bialkowski, Appl. Spectrosc. 63, 815 (2009).
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122
CHAPTER 7
PULSED-LASER EXCITED THERMAL LENS SPECTROSCOPY WITH SAMPLEFLUID HEAT COUPLING c
ABSTRACT
Analytical and finite element analysis modeling methods of the pulsed laser
excited photothermal lens signal of solids samples surrounded by air are presented. The
analytical and finite element analysis solutions for the temperatures induced in the sample
and in the air were found to agree for over the range of conditions in this report. Model
results show that the air contribution to the total photothermal lens signal is significant in
many cases. In fact, these solutions open up the possibility of applying the pulsed excited
TL method for accurate prediction of the heat transfer to the coupling fluid and
subsequently to study the gas surrounding the samples by using a known material solid
sample.
INTRODUCTION
Photothermal (PT) effects are employed in several high-sensitivity measurement
techniques used for material characterization.1-3 Basically, light-induced heating of a
sample produces the PT effect. This heating may cause a number of different effects in
solid, liquid and gas samples.
______________________________________________________________________
c
COAUTHORED BY PR JOSHI, NGC ASTRATH, LC MALACARNE, GVB LUKASIEVICZ, MP
BELANCON, ML BAESSO, AND SE BIALKOWSKI. REPRODUCED WITH PERMISSION FROM
THE JOURNAL OF APPLIED PHYSICS, 107, 083512 (2010) COPYRIGHT 2010 AMERICAN
INSTITUTE OF PHYSICS (SEE APPENDIX C)
123
For instance, the induced temperature change: generates acoustic waves inside the
sample which propagate out to the surroundings; 1 induces increased infrared emission; 1
changes the refractive index of the sample; 4-9 and creates surface deformation.10-13
Information on the temperature rise in the sample as well as its thermo-physical
parameters can be obtained with the different detection methods. 1-16 The limits of
detection for the PT methods are related to how realistically the experimental description
can be theoretically modeled. 17,18 Generally, simple and applicable theoretical models
are obtained by introducing modeling approximations that lead to analytical solutions. In
some cases, these approximations can be accounted for by using appropriated
experimental setups. However, it is not always feasible. In most practical situations, for
example, heat is transferred from the sample to the surroundings along the axial
dimension. This is especially true when the heated sample is in contact with the air or
another coupling fluid. Recently, it has been showed that sample/air heat coupling in the
Thermal Lens (TL) experiment could significantly contribute to the TL signal. 19-21 In
fact, the fluid thermal coupling is always treated as a perturbation to the TL signal. This
perturbation could become stronger as the sample thickness is reduced and/or depending
on the thermo-optical properties of the sample/fluid system.
The theoretical modeling of the thermal lens signal is difficult to be analytically
deducted assuming heat coupling with the surroundings and also adopting realistic
boundary conditions. With approximations, such model can be achieved, but the lack of a
more complete theory makes difficult the validation of the approximated model. In this
sense, finite element methods provide numerical solutions to the heat transfer equations
124
with the realistic boundary conditions imposed by the experimental geometry. Finite
Elemental Analysis (FEA) software has been recently 19-21 applied to describe complex
heat coupling conditions on photothermal techniques. Numerical and approximated
solutions have been compared to the FEA modeling. FEA has shown to be a powerful
tool to model cw and pulsed photothermal methods.
This work presents an analytical theoretical description of the pulsed-laser
excitation thermal lens method by taking the coupling of heat both within the sample and
out to the surroundings into account. Our results are compared with finite elemental
analysis solution, leading to an excellent agreement. The analytical model is then used to
quantify the effect of the heat transfer from the sample surface to the air coupling fluid on
the thermal lens signal. The results showed that the air signal contribution to the total
photothermal lens signal is significant in many cases.
THEORY
The thermal lens effect is based on the heat deposition in a sample by non-radiative
decay process following optical energy absorption by the sample. The sample is heated
by the absorbed optical energy from a pulsed Gaussian profile laser, resulting in a
transverse temperature gradient, which induces a refractive index gradient behaving like
an optical lens. The propagation of another laser beam (the probe beam) through the TL
is affected, resulting in a change in its intensity profile. By measuring the intensity
variation, the information on physical properties of the sample can be obtained.5
The radii of the excitation and probe beams in the sample are 0e and 1P ,
respectively. The probe beam propagates in the z-direction, and the sample is located
125
at z  0 . The sample dimensions are large compared with the excitation beam radius to
avoid edge effects, and the absorbed excitation laser energy by the sample is low so that
the excitation laser can be considered to be uniform along the z-direction.
The TL signal is dependent on the spatially dependent refractive index change
produced due to the temperature change in the sample. Indeed, by heat conduction, a
temperature gradient in the air surrounding the sample is established. One then consider
two semi-infinity spaces with boundaries at z  0 , with the sample in the 0  z   region
and the fluid (air in this work) in the   z  0 region. The temperature rise distributions
inside the sample, Ts  r , z, t  , and in the fluid, T f  r , z, t  , are given by the solution of the
heat conduction differential equations
Ts  r , z , t 
t
 D 2Ts  r , z , t  = Q  r , z 
T f  r , z, t 
t
 D f  2T f  r , z , t  = 0 ,
(1)
(2)
and the boundary and initial conditions are given by
k  z Ts  r , z , t 
z 0
 k f  zT f  r , z, t 
z 0
Ts  , z , t   Ts  r , , t   0
Ts  r , z , 0   T f  r , z, 0   0
Ts  r , 0, t   T f  r , 0, t 
T f  , z , t   T f  r , , t   0
Here Di  ki / i ci is the thermal diffusivity of the material i (sample and fluid). ci , i and
ki
are the specific heat, mass density, and thermal conductivity of the material i ,
respectively. The energy absorbed by the sample, for a collimated, short-pulsed Gaussian
(3)
profile laser propagating on the z-axis direction is Q  r , z, t   Q0 Q  r  Q  z    t  ,
126
with Q  r   exp  2r 2  2  .   t  is the delta function, which is zero unless t  0 .
Q0 = 2 Pe Ae  c  2
and Pe is the excitation beam power, Ae is the optical absorption
coefficient at excitation beam wavelength e , and  is a heat yield parameter accounting
for energy loss due to luminescence during sample excitation. Q  z  = 1  exp   Ae z  is the
absorptance along z-direction. For the purposes here, the optical absorption coefficient is
small and Q  z   1 can be assumed.
Using the integral transform methods, Laplace and Hankel, and the conditions
described in Eq.(3), one can write the solution of the heat conduction differential
equations in the Laplace-Hankel space as


s  D f  2 e  z  s  D  D
Q  
,
Ts  , z , s   

2
2
 k D s  D   s  D
s  D 2 

f
f
2 

 s  D  1 
2 


k
D
s
D
f


Q   k f D k D f
2
(4)
and
T f  , z , s   
Q   s  D f  2
 k D s  D 2
f
2 
 s  D  1  f
k D f s  D 2

e




z
sD  
f
2
Df
s  Df  2
.
(5)
In Esq. (4) and (5), Q    Q0 exp   2 2 8   2 4 and it comes from the Hankel transform of
the source term. For the glass-air system, k f D s  D f  2  k D f s  D 2 , and the term in
1
the denominator of Eqs. (4) and (5) can be expanded in series as 1  x   1  x  x 2  O  x3  .
The temperature gradients are then given by the inverse Laplace and Hankel transforms
127
of Eqs. (4) and (5). Taking the expansion up to the first order, that is,
1  k
f
D s  Df  2 k Df
s  D 2
 , the temperature gradient in the sample, T
1 s 

T1 s   r , z, t    Ts  , z , t   J 0  r  d
0
 r , z, t  , is
,
(6)
with
Ts  , z, t  = Q   e

erf

 Dt 2
  D  D f
 D  D 
f
1/ 2

  2 D f
2
e
 Q  
 e  D 

0
k Df
 

kf D
t


z2
exp  


2
D
t

4
(
)


 d
 e  D (t  )

 (t   )

(7)
For the fluid, the first order expansion, T1 f   r , z, t  , is

T1 f   r , z, t    T f  , z, t   J 0  r  d
(8)
0
with
  2 D f
2
e
T f  , z , t  = Q    
 e  D  erf
0 
 

t
 k D e  D 2 1  2 D 2  2 2 D
f
f

 k Df
 


  D  D f


D  Df  


  e D f  2 (t  ) e z2 4 D f (t  ) d

 (t   )

J n  x  is the Bessel function of the first kind and erf  x  is the error function.
The strength of the photothermal lens element is found from the second radial
derivative, evaluated on-axis. Integration over the path length in the sample and in the
surrounding air results in the inverse focal length 15
(9)
128
1
 dn 


f (t )  dT  f

2
d
T ( r , z , t ) dz 
dr 2 1 f 
r 0
path
d2
 ds 

  path 2 T1 s  ( r , z , t ) dz
dr
 dT  s
r 0
(10)
As J 0  r  0   1 , we can rewrite Eq. 10 as
1
1
1



f (t ) f f (t ) f s (t )
 dn 
2

 dT  f
  T  , z , t    
0


0
 ds  L / 2
 2
 
 dT  s 0
2
2   d dz 
2
2   d dz
f
 T  , z , t    

0
s
(10)
The space and time dependences of the temperature equations for the air and
samples media, Eqs. (6) to (9), and the inverse focal length, Eq. (10), will be compared
to the FEA modeling solutions for a system with glass sample of thickness l surrounded
by air. The numerical integrations in these equations are performed by using standard
command in the software Mathematica (version 7.0).
To compare the analytical solutions presented previously, FEA software was used
to perform numerical solutions to the heat transfer equations with the realistic boundary
conditions. Comsol Multiphysics 3.5 analysis is carried out on a Dell Studio XPS 435, i7
940 processor, using MS Windows Vista. The Comsol Multiphysics software in
conduction and convection mode solves the heat diffusion equation given as
c
T  r , z, t 
t
 k  2T  r , z , t  =  cQ  r , z , t    c u  T  r , z , t 
in which u is the flow velocity. All other symbols are the same as those introduced
above. Note that Eqs. (1), (2) and (11) differ only by the second term on the right side of
the equation. This term can account for convection or mass flow heat transfer, which is
not important for the sample investigated in this work: glass.
(11)
A detailed description of FEA modeling can be found elsewhere.
16,20,21
129
Briefly, it consists of drawing the sample geometry and specifying material boundary
conditions, heat sources, and sinks. The problems are then solved with rough finite
element definition and further refinement of elements and domain are made. The element
mesh is refined until model results become independent of mesh size. Finally, the
temperature profile can be obtained either at a single time, over a time series, or at steady
state, for the sample and air domain. The model solid absorbing media was cylindrical
plate of 10 mm in diameter and 1mm thick. Optical excitation was along z-axis.
The values of the thermal, optical and mechanical parameters used for the
analytical and FEA modeling simulations are shown in Table 7-1.
Table 7-1. Parameters used for the simulations. The thermal, optical and mechanical
properties listed below are associated to characteristics values found in glasses 13and for
the surrounding medium we use the air properties. 22
Parameters
(units)
Glass sample Air
D
(10-7m2/s)
5.0
2.2
k
(W/mK)
1.4
0.026

(kg/m3)
933
1.18
c
(J/kgK)
3000
1005
ds dT
(10-6 K-1)
10.0
-1.0

(m)
50
tc   2 4 D
(ms)
1.25
130
RESULTS AND DISCUSSION
Figure 7-1 shows the normalized radial temperature, T1 s   r , 0.5, t  / T1 s   r , 0.5, 0  within
the glass sample at its center, z  0.5 mm . The results by the FEA model are compared to
the temperature solution considering heat flux from glass to air, Eq. (6) at different
exposure times. One can see that there is almost no difference between the FEA model
and the semi-infinite approximation with glass-air heat coupling.
Figure 7-2 shows the normalized temperature profiles along the z direction, from
air to glass domain, using the FEA modeling, the solution considering air surroundings,
Eq. (6), and the solution with no transfer of heat from glass to air, Eq. (8), at different
exposure times at r  0 . It clearly shows an excellent agreement between the analytical
solutions considering air coupling and the FEA model.
Figure 7-1. Normalized temperature profile in the glass sample at z  0.5 mm using the
FEA modeling, the solution considering heat flux to air, Eq. (6) at different exposure
times.
131
Figure 7-2. Normalized temperature profile along the z direction (air-glass) using the
FEA modeling and the solution considering air surroundings, Eqs. (6) and (8), at different
exposure times. r  0 was used in the simulations.
As displayed by Fig.7-3, the agreement between both predictions is also very good
for the normalized radial temperature profile in the air surroundings using the FEA
modeling and the solution for the air fluid, Eq. (8), at the sample surface on the air side,
z 0,
and up to 0.2mm distant. The exposure time was t  1ms .
The results obtained using FEA model show that the first order analytical
approximation presented in this work can be used to describe the temperature profile in
the sample and in the air surrounding it.
132
Figure 7-3. Radial temperature profile in the air surroundings at t  1ms using the FEA
modeling and the solution for the air fluid, Eq. (8), at the sample surface in the air side,
z 0.
Figure 7-4. Normalized radial temperature profile in the sample and in the air
surroundings at t  0.5ms using Eqs. (6) and (8).
133
Figure 7-4 shows a complete picture of the normalized radial temperature
gradient along the z-direction at t  0.5ms .
The solutions presented in Eqs. (6) and (8) can be analytically used to describe
theoretically the thermal lens effect, by calculating the inverse focal length of the probe
beam after passing thought the TL region, Fig. 7-5. The air and samples absolute
contributions are normalized by the sample f s  0 1 inverse focal length at t  0 . The
results are then compared to the FEA modeling. For the FEA solutions for the inverse
focal length, one performed the integration over the second radial derivative of the
temperature change evaluated on r  0 .
(open sym bols) FE A solution
(continuous lines) A nalytical solution
Sam ple
1.0
1.0
0.8
-1
-1
|fs,f(t) /fs(0) |
0.6
0.6
Sam ple
0.4
Air
0.002
-1
-1
|fs,f(t) /fs(0) |
0.8
0.4
0.000
0.0
0.2
0.1
0.2
0.3
0.4
Tim e (m s)
Air
0.0
0
1
2
3
4
5
6
7
T im e (m s)
Figure 7-5. FEA and analytical solution using Eq. (10) for the normalized inverse focal
length for the 1mm thick sample. We used  dn / dT  f  1 10 6 K 1
and  ds / dT s  10 106 K 1 .
134
The time dependent thermal lens signal calculated using FEA and the
analytical solutions are in good agreement.  ds / dT s  10 106 K 1 was used for the glass
sample and  dn / dT  f  1 10 6 K 1 for the air. For a relatively thick sample, 1mm, both the
FEA and analytical solutions for the inverse focal length showed no significant
difference. Fig. 7-5 shows that the magnitude of the signal of glass is more than three
orders greater than that of the air. This is because the temperature change with pulsed
laser excitation is much faster than the heat diffusion to the surroundings. However, this
result is highly dependent on the thermo-optical parameters of the sample and heat
coupling fluid and the sample thickness. It can be exemplified by performing a relative
change of the air and sample signals for different sample thickness. Fig.7- 6 shows the
absolute ratio between the air and sample signals in function of the sample thickness.
Figure 7-6. Absolute ratio between the air and sample signal, f  t f 1 f  t s 1 , in function
of the sample thickness with  dn / dT  f  1 106 K 1 and  ds / dT s  10 106 K 1 .
135
From Fig.7-6 one can see that the effect of the air on the sample signal is very
small for thick samples. On the other hand, as the thickness decreases, the air
photothermal signal becomes stronger and it has a significant contribution to the total
inverse focal length. In fact, the air signal contribution is about 6% of the sample signal.
It is important to note that the air signal is a negative contribution to the total inverse
focal length,  dn / dT  f  0 .
This behavior is also observed even with thick samples when  dn / dT  f presents
values of the same order of  ds / dT s . Experimentally, it can be obtained by using a heat
coupling fluid with higher  dn / dT  f an/or by using different semitransparent samples
with smaller  ds / dT s . To illustrate this, Fig. 7- 7 shows the absolute ratio between the air
and sample signals in function of the  dn / dT  f  ds / dT s ratio. If the absolute value of
 dn / dT  f is small compared to  ds / dT s , one have the case considered previously, where
the effect of the air on the sample signal is very small. However, if they become
comparable or if  dn / dT  f   ds / dT s , the effect of the air on the photothermal signal
ensues significant.
Figure 7-8 shows the normalized inverse focal length for a typical glass-air
system in function of the sample thickness.  dn / dT  f  1 106 K 1 and
 ds / dT s  10 106 K 1 were used in the numerical calculations using Eq. (10). For thin
samples, there is an initial rapid decrease in inverse focal length.
136
Figure 7-7. Absolute ratio between the air and sample signal, f  t f 1 f  t s 1 , in function
of the  dn / dT  f  ds / dT s ratio.
Figure 7-8. Analytical solution using Eq. (10) for the normalized inverse focal length as
a function of the sample thickness with  dn / dT  f  1 10 6 K 1 and  ds / dT s  10 106 K 1 .
137
This rapid decay is due to heat loss to the air at the surfaces, which decreases the
length of the photothermal lens. This effect is less pronounced as the thickness increases
due to the fact that border effects are less important for thick samples. The fact that the
air signal contribution to the inverse focal length becoming significant for some cases, for
instance, when the sample thickness is reduced and/or  dn / dT  f is comparable
to  ds / dT s , or  dn / dT  f   ds / dT s , turns out to be an interesting point for this technique.
It could be used to determine thermo-optical properties of the surrounding media
providing the physical parameters of the sample are known.
CONCLUSION
Analytical and finite element analysis modeling were used to describe the pulsed
laser excited photothermal lens signal by considering both the heat transfer from sample
to air and the TL generated in the air surroundings. The analytical solutions for the
temperatures induced in the sample and in the air were found to agree with that of the
FEA software. The results showed that the air signal contribution to the total
photothermal lens signal is significant in many cases, mainly for those where the absolute
value of  dn / dT  f is comparable to that of  ds / dT s and also  dn / dT  f   ds / dT s . In
fact, these solutions open up the possibility of applying the pulsed excited TL method for
accurate prediction of the heat transfer to the coupling fluid and subsequently to study the
gas surrounding the samples by using a known material solid sample. Finally, on one
side, the analytical solutions presented here are in complete agreement with the FEA
modeling and it could be used for material parameter determination. On the other hand,
138
FEA modeling opens up numerous possibilities of analyzing theoretically different
samples at different external conditions.
REFERENCES
1.
A. Mandelis, ed., Progress in Photoacoustic and Photothermal Science and
Technology (Elsevier, New York, 1991).
2.
S. E. Bialkowski, Photothermal Spectroscopy Methods for Chemical Analysis
(Wiley, New York, 1996).
3.
D. P. Almond, and P. M. Patel, Photothermal Science and Techniques (Chapman
& Hall, London, 1996).
4.
J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, J.
Appl. Phys. 36, 3 (1965).
5.
J. Shen, R. D. Lowe, and R. D. Snook, Chem Phys. 165, 385 (1992).
6.
M. L. Baesso, J. Shen, and R. D. Snook, J. Appl. Phys. 75, 3732 (1994).
7.
M. L. Baesso, A. C. Bento, A. A. Andrade, J. A. Sampaio, E. Pecoraro, L. A. O.
Nunes, T. Catunda, and S. Gama, Phys. Rev. B 57, 10545 (1998).
8.
S. M. Lima, T. Catunda, R. Lebullenger, A. C. Hernandes, M. L. Baesso, A. C.
Bento, and L. C. M. Miranda, Phys. Rev. B 60, 15173 (1999).
9.
N. G. C. Astrath, J. H. Rohling, A. N. Medina, A. C. Bento, M. L. Baesso, C.
Jacinto, T. Catunda, S. M. Lima, F. G. Gandra, M. J. V. Bell, and V. Anjos, Phys.
Rev. B 71, 14202 (2005).
10.
N. G. C. Astrath, L. C. Malacarne, P. R. B. Pedreira, A. C. Bento, and M. L.
Baesso, and J. Shen, Appl. Phys. Lett. 91, 191908 (2007).
139
11.
L. C. Malacarne, F. Sato, P. R. B. Pedreira, A. C. Bento, R. S. Mendes, M. L.
Baesso, N. G. C. Astrath, and J. Shen, Appl. Phys. Lett. 92, 131903 (2008).
12.
F. Sato, L. C. Malacarne, P. R. B. Pedreira, M. P. Belancon, R. S. Mendes, M. L.
Baesso, N. G. C. Astrath, and J. Shen, J. Appl. Phys. 104, 053520 (2008).
13.
N. G. C. Astrath, F. B. G. Astrath, J. Shen, J. Zhou, P. R. B. Pedreira, L. C.
Malacarne, A. C. Bento, and M. L. Baesso, Opt. Lett. 33, 1464 (2008).
14.
A. Marcano, O. L. Rodriguez, and Y. Alvarado, J. Opt. A: Pure Appl. Opt. 5,
S256 (2003).
15.
W. B. Jackson, N. M. Amer, A. C. Boccara, and D. Fournier, Appl. Opt. 20,1334
(1981).
16.
O. O. Dada, M. R. Jorgensen, and S. E. Bialkowski, Appl. Spectrosc. 61, 1373
(2007).
17.
A. Chartier, and S. E. Bialkowski, Opt. Eng. 36, 303 (1997).
18.
J. R. Whinnery, Acc. Chem. Res. 7, 225 (1974).
19.
L. C. Malacarne, N. G. C. Astrath, P. R. B. Pedreira, R. S. Mendes, M. L. Baesso,
P. R. Joshi, and S. E. Bialkowski, Submitted to J. Appl. Phys. (2009).
20.
O. O. Dada, and S. E. Bialkowski, Appl. Spectrosc. 62, 1326 (2008).
21.
P. R. Joshi, O. O. Dada, and S. E. Bialkowski, Appl. Spectrosc. 63, 815 (2009).
22.
D. R. Lide, CRC Handbook of Chemistry and Physics (88th Ed., CRC Press,
Cleveland, 1977).
140
CHAPTER 8
SUMMARY
The research work outlined in this dissertation focuses on pulse-laser excited
photothermal lensing effect on glasses and a small size cylindrical sample cell. The
COMSOL multiphysics finite element analysis software modeling of the pulse-laser
excited glass shows that there is: (1) heat transfer between the glass surface and the air
coupling fluid, (2) fast heat transfer along the axial direction which is the direction of
propagation of the excitation source, and (3) glass surface deformation due to pulse-laser
heating. These issues have been addressed for the application of photothermal lens
spectroscopy to the study of glass correctly.
First, finite elemental analysis software is used to model the photothermal effect by
simulating the coupling of heat both within the sample and out to the surroundings.
Modeling shows that there is significant amount heat transfer from the glass surface to
the air. And at the same time the air signal contribution to the total photothermal lens
signal is significant. Although heat transfer between the glass surface and the air coupling
fluid has a significant effect on the predicted time dependent photothermal lens signals, a
pulse-laser excitation has been found to be a better option in comparison to continuous
laser excitation for photothermal lens spectroscopy of glass surrounded by air. This is
because of the slower heat transfer from the glass to the air surrounding than the time
dependent photothermal lens signals decay in sample for pulse-laser excitation.
For comparison with experimental signals, a simple equation based on the finite element
analysis result is proposed for accounting for the variance of experimental data where this
141
type of heat coupling situation occurs. Experimental results for photothermal lens
measurements are compared to finite elemental analysis models for commercial colored
glass filters. The colored glass filters are found to have positive thermo-optical
coefficients and considered to be the consequence of counteracting factors: optical
nonlinearity, stress-induced birefringence, and the structural network of glass.
Secondly, the fast initial signal decay in glass, which is shown by the FEA
modeling of photothermal lens signal of colored silica glass surrounded by air, is
interpreted as due to the diffusion of heat from the surface of glass. This is because when
the heated sample is direct in contact with air or another coupling fluid, heat is transferred
from sample to the surroundings along to the axial dimension. And this problem is more
severe, particularly when the sample is thin and insulated. The fast signal decay is
detected by using fast response detector in transmission and reflection photothermal lens
experiments. FEA modeling is used to evaluate the time constant of slow and fast
components and corrections are made for the prediction of material properties.
Finite heat transfer at the interface is typically neglected in semi-infinite cylinder
approximations. In order to address the issue of heat coupling between the sample and its
immediate surrounding an analytical and finite element analysis modeling were used to
describe the pulsed and continuous laser excited photothermal lens signal by considering
both the heat transfer from sample to air and the thermal lens generated in the air
surroundings. The analytical solutions for the temperatures induced in the sample and in
the air were found to agree with that of the FEA modeling. The results showed that the air
signal contribution to the total photothermal lens signal is significant in many cases,
142
mainly for those where the absolute value of  dn / dT  f is comparable to that of
 ds / dT s and also  dn / dT  f
  ds / dT  s .
Third, The FEA modeling shows that there is surface deformation due to pulse
laser heating of colored glass filters, and consequently the optical path length of them
changed. The modeling also shows that the extent of deformation increases with
absorption coefficient of glass. It has been found that the photothermal lens signals and
thermal expansion are affected due to air environment around the solid sample. The
surface deformation phenomenon can be exploited for the measurement of thermal and
physical properties of solid materials. In short, FEA modeling can be used to better
understand the pulsed laser excited photothermal spectroscopy signals by correctly
accounting for the heat transfer of the sample to the surroundings where the absorbing
material is in direct contact with the coupling fluid surroundings. The cylindrical
approximation is not valid in these cases.
The fluid thermal coupling is treated as a perturbation to the thermal lens signal.
FEA modeling could be used to correct the perturbation to the thermal lens signal for
material parameter determination. These studies open up the new avenues of application
of FEA modeling and the pulsed excited thermal lens method for accurate prediction of
the heat transfer to the coupling fluid and subsequently to study the gas surrounding the
samples by using a known material solid sample. However, a comparative study of
different materials of different optical and thermal properties is required with the help of
these methods for the best results.
143
Lastly, realizing the advantages of cylindrical sample cell over conventional
cell, a novel apparatus for performing photothermal lens spectroscopy with pulsed
excitation laser is described. When the whole sample cell volume is irradiated with
constant irradiance beam produced by an excitation laser, the photothermal lens element
is formed by thermal diffusion from the irradiated sample volume through the sample cell
walls. The parabolic-form photothermal lens produced by thermal diffusion is aberration
free. A low-volume cylindrical sample cell requires a little sample and signals produced
by thermal diffusion through the sample wall are less susceptible to the bulk heat transfer
effects due to convection. The apparatus is found to work with cells designed to contain
sample volumes to nL level.
FEA modeling is used to examine the temperature profile and the photothermal
signal and result of it is compared with the experimental result. The experimental
photothermal lens enhancement is found to be that predicted from theory within
experimental error. A best advantage of cylindrical sample cell is that it circumvents the
large temperature changes by heat transport to the surrounding. However, further study to
optimize the apparatus and the experiment is required for the best performance.
144
APPENDIXES
145
APPENDIX A:
Figure and Data
All figure data are stores in the "MGD" media device that accompanies this dissertation.
146
APPENDIX B:
Comsol Multiphysics Files
Comsol Multiphysics files for finite element analysis modeling results are stored in
"MGD" media device that accompanies this dissertation.
147
APPENDIX C:
Permission letters
148
149
From: RIGHTS <[email protected]>
Date: Wed, Apr 7, 2010 at 10:13 AM
Subject: Re: Fwd: copyright permission
To: [email protected]
Dear Dr. Joshi:
Thank you for requesting permission to reproduce material from American
Institute of Physics publications.
Permission is granted – subject to the conditions outlined below – for the
following:
“Analytical solution for mode-mismatched thermal lens spectroscopy with
sample-fluid heat coupling,”
Journal of Applied Physics Volume 107, Issue 5, 053104 (2010).
To be used in the following manner:
Reproduced in your dissertation to be submitted to the Department of Chemistry
and Biochemistry at Utah State University.
The American Institute of Physics grants you the right to reproduce the
material indicated above on a one-time, non-exclusive basis, solely for the
purpose described. Permission must be requested separately for any future or
additional use.
Please let us know if you have any questions.
Sincerely,
Susann Brailey
~~~~~~~~~~~
Office of the Publisher, Journals and Technical Publications
Rights & Permissions
American Institute of Physics
Suite 1NO1
2 Huntington Quadrangle
Melville, NY 11747-4502
516-576-2268 TEL
516-576-2450 FAX
150
from: RIGHTS <[email protected]>
to: Prakash Joshi <[email protected]>
date: Mon, May 17, 2010 at 12:04 PM
subject: Re: Permission letter
mailed-byaip.org
Dear Dr. Joshi:
Thank you for requesting permission to reproduce material from American
Institute of Physics publications.
Permission is granted – subject to the conditions outlined below – for the
following:
Journal of Applied Physics, Volume 107, Issue 8, 083512 (2010).
To be used in the following manner:
Reproduced in your dissertation for submission to the department of chemistry
and biochemistry at Utah State University.
The American Institute of Physics grants you the right to reproduce the
material indicated above on a one-time, non-exclusive basis, solely for the
purpose described. Permission must be requested separately for any future or
additional use.
Please let us know if you have any questions.
Sincerely,
Susann Brailey
~~~~~~~~~~~
Office of the Publisher, Journals and Technical Publications
Rights & Permissions
American Institute of Physics
Suite 1NO1
2 Huntington Quadrangle
Melville, NY 11747-4502
516-576-2268 TEL
151
516-576-2450 FAX
[email protected]
from: Gustavo Lukasievicz <[email protected]>
to: [email protected]
date: Wed, Apr 28, 2010 at 9:14 PM
subject: Permission to reprint the manuscripts
mailed-byhotmail.com
------------------------------------------------------------------------------Dear Prakash,
I hereby give my permission to Prakash Raj Joshi to reprint the manuscript
entitled “Pulsed-Laser Excited Thermal Lens Spectroscopy With Sample-Fluid Heat
Coupling” co-authored by Nelson G. C.Astrath, Luis C. Malacarne, Gustavo V. B.
Lukasievicz, Marcos P. Belancon, Mauro L. Baesso, Prakash R. Joshi, and Stephen
E.Bialkowski
Regards,
Gustavo
---------------------------------------Gustavo V. B. Lukasievicz
Universidade Estadual de Maringá
Av. Colombo 5790, CEP 87020-900
Maringá - Paraná - Brasil
----------------------------------------
152
from: [email protected]
to: Prakash Joshi <[email protected]>
date: Tue, Apr 20, 2010 at 6:58 AM
subject: Re: Permission letter request
mailed-by: dfi.uem.br
Dear
Prakash Raj Joshi
I hereby give my permission to Prakash Raj Joshi to reprint the manuscripts
entitled “Analytical Solution For Mode-Mismatched Thermal Lens Spectroscopy
with Sample-Fluid Heat Coupling”, co-authored by Luis C. Malacarne, Nelson
G. C. Astrath, Paulo R. B. Pedreira, Mauro L. Baesso, Renio S. Mendes,
Prakash R. Joshi and Stephen E. Bialkowski and “Pulsed-Laser Excited Thermal
Lens Spectroscopy With Sample-Fluid Heat Coupling” co-authored by Nelson G.
C. Astrath, Luis C. Malacarne, Gustavo V. B. Lukasievicz, Mauro L. Baesso, Marcos
P. Belancon, Prakash R. Joshi, and Stephen E.Bialkowski
Dr. Luis C. Malacarne
Departamento de Física
Universidade Estadual de Maringá
Maringá-PR, 87020-900, Brazil
153
from :Marcos Paulo Belançon <[email protected]>
to :[email protected]
date :Thu, Apr 15, 2010 at 1:13 AM
subject Fwd: Permission letter
mailed-bygmail.com
signed-bygmail.com
-----------------------------------------------------------------------------------Dear Prakash,
I hereby give my permission to Prakash Raj Joshi to reprint the manuscript entitled
“Pulsed-Laser Excited Thermal Lens Spectroscopy With Sample-Fluid Heat Coupling”
co-authored by Nelson G. C. Astrath, Luis C. Malacarne, Gustavo V. B. Lukasievicz,
Mauro L. Baesso, Marcos P. Belancon, Prakash R. Joshi, and Stephen E.Bialkowski
Regards,
---------------------------------------------------------------Marcos Paulo Belançon
Universidade Estadual de Maringá
Departamento de Física
Av. Colombo 5790, CEP 87020-900
Fone: +55(44)3261-4330
Maringá - Paraná - Brasil
---------------------------------------------------------------Marcos Paulo Belançon
Universidade Estadual de Maringá
Curso de Doutorado em Física
Université Claude Bernard Lyon I
LPCML - (+33) 04 72 43 29 71
Celular - (+33) 0633220268
154
from: Mauro Luciano Baesso (UEM) <[email protected]>
to: [email protected]
date:Thu, Apr 15, 2010 at 4:50 AM
subject ENC: Permission letter
mailed-byuem.br
-----------------------------------------------------------------------------------Dear Prakash,
I hereby give my permission to Prakash Raj Joshi to reprint the manuscripts entitled
“Analytical Solution For Mode-Mismatched Thermal Lens Spectroscopy with SampleFluid Heat Coupling”, co-authored by Luis C. Malacarne, Nelson G. C. Astrath, Paulo R.
B. Pedreira, Mauro L. Baesso, Renio S. Mendes, Prakash R. Joshi and Stephen E.
Bialkowski and “Pulsed-Laser Excited Thermal Lens Spectroscopy With Sample-Fluid
Heat Coupling” co-authored by Nelson G. C. Astrath, Luis C. Malacarne, Gustavo V. B.
Lukasievicz, Mauro L. Baesso, Marcos P. Belancon, Prakash R. Joshi, and Stephen
E.Bialkowski
Best regards,
Mauro L Baesso
---------------------------------------------------------------XXXX
Universidade Estadual de Maringá
Departamento de Física
Av. Colombo 5790, CEP 87020-900
Fone: +55(44)3261-4330
Maringá - Paraná - Brasil
155
from :Nelson G. C. Astrath <[email protected]>
to :Prakash Joshi <[email protected]>
date :Wed, Apr 14, 2010 at 9:53 PM
subject :Permission letter
mailed-bypq.cnpq.br
Dear Prakash,
I hereby give my permission to Prakash Raj Joshi to reprint the manuscripts
entitled “Analytical Solution For Mode-Mismatched Thermal Lens Spectroscopy
with Sample-Fluid Heat Coupling”, co-authored by Luis C. Malacarne, Nelson G.
C. Astrath, Paulo R. B. Pedreira, Mauro L. Baesso, Renio S. Mendes,Prakash R.
Joshiand Stephen E. Bialkowski and“Pulsed-Laser Excited Thermal Lens
Spectroscopy With Sample-Fluid Heat Coupling” co-authored by Nelson G. C.
Astrath, Luis C. Malacarne, Gustavo V. B. Lukasievicz, Mauro L. Baesso, Marcos P.
Belancon, Prakash R. Joshi, and Stephen E.Bialkowski
Regards,
Nelson
---------------------------------------Nelson Guilherme Castelli Astrath
Universidade Estadual de Maringá
Av. Colombo 5790, CEP 87020-900
Fone: +55(44)3041-5910,+55(44)9944-9570
Maringá - Paraná - Brasil
----------------------------------------
156
from :Paulo Roberto Borba Pedreira <[email protected]>
to :[email protected]
date :Wed, Apr 14, 2010 at 10:53 PM
subject :Permission to reprint
mailed-bygmail.com
signed-bygmail.com
Dear Prakash,
I hereby give my permission to Prakash Raj Joshi to reprint the manuscript entitled
“Analytical Solution For Mode-Mismatched Thermal Lens Spectroscopy with SampleFluid Heat Coupling”, co-authored by Luis C. Malacarne, Nelson G. C. Astrath, Paulo R.
B. Pedreira, Mauro L. Baesso, Renio S. Mendes, Prakash R. Joshi, and Stephen E.
Regards,
---------------------------------------------------------------Paulo Roberto Borba Pedreira
Universidade Estadual de Maringá
Departamento de Física
Av. Colombo 5790, CEP 87020-900
Fone: +55(44)3261-4330
Maringá - Paraná - Brasil
----------------------------------------------------------------
157
from; Renio dos Santos Mendes <[email protected]>
to: [email protected]
date: Thu, Apr 22, 2010 at 5:12 AM
subject: Permission letter]
mailed-by: dfi.uem.br
Dear Prakash,
I hereby give my permission to Prakash Raj Joshi to reprint the manuscripts entitled
"Analytical Solution For Mode-Mismatched Thermal Lens Spectroscopy with SampleFluid Heat Coupling", co-authored by Luis C. Malacarne, Nelson G. C. Astrath, Paulo R.
B. Pedreira, Mauro L. Baesso, Renio S. Mendes, Prakash R. Joshi, and Stephen E.
Bialkowski.
Regards,
---------------------------------------------------------------Renio dos Santos Mendes
Universidade Estadual de Maringa
Departamento de Fi-sica
Av. Colombo 5790, CEP 87020-900
Fone: +55(44)3261-4330
Maringa - Parana - Brasil
158
CURRICULUM VITAE
Prakash Raj Joshi
Utah State University
Department of Chemistry and Biochemistry
Logan,Utah. 84322-0300
PROFESSIONAL STATUS: PhD Analytical Chemistry
PHONE : 435-797-6278
E-MAIL: [email protected]
: [email protected]
AREAS OF INTEREST
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Photothermal spectroscopy
Synthesis and characterization of nanoparticles
Raman, Fluorescence, IR Absorption and Emission Spectroscopy
GC and HPLC
Atomic Absorption and Emission Spectroscopy
EDUCATION
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Ph.D., Analytical Chemistry, Utah State University, US (2010)
Dissertation: Pulsed-Laser Exited Photothermal Study of Glasses and Nanoliter
Cylindrical Sample Cell Based on Thermal Lens Spectroscopy
M. Phil., Physical Chemistry, Kathmandu University, Nepal (1999)
Dissertation: Synthesis, Characterization, and thermal studies of complexes of
Copper (II), Mercury (II) and Cobalt (II) with sulfur and Nitrogen containing
ligands.
M. Sc., Physical Chemistry, Tribhuwan University, Kathmandu, Nepal (1994)
Dissertation: Spectroscopic studies on Methylineblue crystal violet and complexes
with Montmorillonite.
B. Sc., (Major: Chemistry, Zoology & Botany) Tribhuwan University,
Kathmandu, Nepal (1989)
159
PROFESSIONAL EXPERIENCE
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Graduate research and teaching assistant, Department of Chemistry and
Biochemistry, Utah State University (2005-2010)
In-charge, Headed the department, Chemistry Department, Kathmandu University
(2002-2005)
Assistant professor, Kathmandu University, Nepal (2000-2005)
Lecturer, Chemistry Department, Kathmandu University (1994-2000)
Instructor, Chemistry Department, Kathmandu University (1993-1994)
High School Science Teacher, Trinagar High School, Nepal (1989-1990)
Member, Chemistry Subject Committee, Kathmandu University (1994-2002)
Chairman, Chemistry Subject Committee, Kathmandu University (2002-2005)
In-charge, Instrumentation Laboratory, Chemistry Department, Kathmandu
University (2002-2005)
Member, Faculty-Recruiting Committee, Chemistry Department, Kathmandu
University (2002-2005)
Expert Member of Arsenic Monitor for Water, Royal Nepal Academy of Science
and Technology, RONAST (2002-2003)
COMPUTATIONAL SKILLS
Comsol Multiphysics (Finite element analysis modeling software)
Code V (Optical design software)
PROFESSIONAL SOCIETY AFFILIATIONS
American Chemical Society
Society for Applied Spectroscopy
AWARDS
National Education Award, 2002, Education Ministry, His Majesty Government of Nepal
160
PUBLICATION
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Prakash R. Joshi,Oluwatosin O. Dada, and Stephen E. Bialkowski “Pulsed Laser
Excited Photothermal Lens Spectrometry of CdSxSe1-x Doped Silica Glasses”
Applied Spectroscopy, 63 (7) 815-821(2009)
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Prakash R. Joshi, Oluwatosin O. Dada, and Stephen E. Bialkowski “Pulsed Laser
Excited Photothermal Lens Spectrometry of Cadmium Sulfoselenide Doped Silica
Glasses” Journal of Physics: Conference Series 214 (2010) 012117

Luis C. Malacarne, Nelson G. C. Astrath, Paulo R. B. Pedreira, Renio S. Mendes,
Mauro L. Baesso, Prakash R. Joshi, and Stephen E. Bialkowski “Analytical
solution for mode-mismatched Thermal Lens spectroscopy with sample-fluid heat
coupling” Journal of Applied Physics, 107 , 053104 ( 2010)

Nelson G. C. Astrath, Luis C. Malacarne, Gustavo V. B. Lukasievicz, Marcos P.
Belancon, Mauro L. Baesso,Prakash R. Joshi, and Stephen E. Bialkowski,
“pulsed-laser excited thermal lens spectroscopy with sample-fluid heat coupling”
Journal of Applied Physics 107, 083512 (2010)
SCIENTIFIC MEETING AND PRESENTATIONS
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“Pulsed Laser Excited Photothermal Lens Spectrometry of CdSxSe1-x Doped
Silica Glasses”, Prakash R. Joshi Oluwatosin O. Dada, and Stephen E.
Bialkowski. 237th American Chemical Society, ACS, National Meeting &
Exposition March 22-26, 2009 Salt Lake City, UT

“Photothermal Lens Spectrometry in Nanoliter Cylindrical Sample Cells” Prakash
R. Joshi, and Stephen E. Bialkowski 238th American Chemical Society, ACS,
National Meeting & Exposition August 16-20, 2009 Washington, DC